lesser-known things in Florence

There have been museum exhibitions in the park Belvedere above the Pitti Palace. Pre-Covid, there was a major music/dance performance festival called Maggio Musicale.

I'm sure these are very hard to find/look up, but we were lucky enough to run across (years ago) weekly (government run) bus tours that entered at least the gardens of several private Palazzi in the countryside, like Harvard's. Only one or two of them let the group partway inside, but it was still special. We either found a brochure for it in a tourist rack, or learned at the TI. We actually trained back to Florence from Reggio Emilia to take the tour. (Besides the excellent food in Reggio, there's a nice little town museum with a few Emilian paintings, and a replica full-scale baroque theater that has tours. Market day might still be Saturday.)

I have this bookmarked
https://www.ourescapeclause.com/things-to-do-in-florence/

Santa Trinita, Sassetti Chapel (Ghirlandaio)*

Bardini museum and gardens*

Torre di San Niccolo

Flower market Thursday mornings Piazza della Repubblica*

Forte di Belvedere

Museo Archeologico *

Museo dell’Opera del Duomo *

Museo Horne

Museo Opificio delle Pietra Dure

Museo Stibbert and garden

Palazzo Medici-Riccardi (Gozzoli chapel)*

Parco della Cascine

San Lorenzo *

San Miniato al Monte and cemetery*

Santa Felícita (Pontormo)*

Ospedale degli Innocenti *

day trip to Pistoia *

Santa Maria Novella pharmacy

Laurentian library

take an art history class at the British Institute*

(The ones with asterisks are things I have greatly enjoyed myself, the others are on my next-time list.)

The Ferragamo Shoe Museum!
I happened upon it one year when it was baking hot out and I needed to get into some AC.
I was in there for a couple of hours, and it's really interesting, as well as nice and cool on a hot day.

Also, the Gucci Museum, also fascinating.

The Bardini Gardens...never seems to be many people there, and much smaller and prettier than the Boboli Gardens.

A Last Supper fresco that a lot of people don't know about:
https://whichmuseum.com/museum/cenacolo-di-fuligno-florence-5874

The Torrigiani Gardens in Oltrarno.
You used to have to apply to get a private viewing, but it might be public now, as they needed the money.
https://www.destinationflorence.com/en/details/656-torrigiani-garden

The Cascine Market on certain days of the week; it's vast and interesting to watch everyone shopping.

Have a look at Girl in Florence, a wonderful resource for the city.
https://girlinflorence.com/

I love Florence too, and have spent a bit of time there.
Hoping to go back in September!

To highlight a few items from nancy's list above:
We particularly enjoyed the Museo Archeologico, and in fact went there a couple of times. Contains the best selection of Roman, Egyptian and Etruscan artifacts and displays we've seen anywhere, and with no crowds to speak of. It's close to the Ospedale degli Innocenti which is just around the corner, and we happened upon a terrific hole in the wall bistro (Un Caffee on via Battisti) about a block away that featured the best paninis we had in the city.
The Bardini Gardens are indeed terrific, and their little cafe might be the best place in the city for lunch, with spectacular views across the river overlooking the city. They whip up great cappuccinos at a reasonable price too. The gardens were only a couple of blocks away from our accommodations so we stopped in several times.
Just below Piazzale Michaelangelo is the very peaceful and scenic Giardino delle Rose. It too features terrific views out over the city, but again without the crowds that all seem to bypass it as they slog further up the hill. You'll likely have it mostly to yourself and the roses should be in full bloom during your May visit. Can grab a panini a couple of blocks down the hill at Come Dio Comanda on Via di S. Niccolo and have a little picnic in the park.
Not exactly in the city proper, but we love the gentle walk/hike from Fiesole along the Belvedere walkway and some forest paths to Monte Ceccere to visit a small monument commemorating Leonardo da Vinci and the first experimental test flight of one of his gliders from the spot back in 1508. There's a picnic table there too if you want to bring a lunch.
For an extended stay in Florence you might be interested in the Amici degli Uffizi card - an annual membership that includes unlimited (and priority) entrance into the Uffizi, Pitti Palace, the Boboli Gardens, Bardini and the Museo Archeologico. We wore ours out when we were there a few years ago. It's a great deal and a terrific money saver.

Edited to add: you can save a lot of wear and tear on your legs if you take advantage of the local buses. Google maps displays the various stops around the city along with a system overview so you can plan your outings in detail. Can buy tickets at any tobacconist shop. Can also just pay the driver when boarding, though it'll cost a little more.
Also, the local professional soccer team is called Fiorentina, and attending one of the team's matches one afternoon might be entertaining if you have the interest. .

For the latest ideas there's an excellent English language publication: https://www.theflorentine.net

We will also be in Florence at the end of May, so I'm bookmarking this discussion - what a wealth of knowledge there is here! Like you, I've been lucky enough to spend a lot of time in Florence, but it never seems like enough. I've definitely added the Museo Archelogico and Bardini Gardens to our list.

For our trip, I also just booked a rafting down the Arno excursion for myself and my five-year old through Firenze Rafting (my work VPN is blocking me from accessing the ".it" web address, or I'd link you to the specific page). It seems like you float down the river in white water rafting boats, but without the rapids, and we're looking forward to seeing the city from a new perspective. I've been communicating with Zeno, the owner/manager, and he's incredibly friendly and helpful.

A few days ago, someone else just posted about a groundbreaking Donatello exhibit that will be in Florence this summer. That post is here, in case you missed it: https://community.ricksteves.com/travel-forum/italy/donatello-exhibit-in-florence I'm not sure that it's necessary, but we also purchased advanced tickets for that.

Cheers to Florence!

Thank you so much to you all for all these great tips and references/resources. Bookmarking them myself.
Nancy

Thanks for sharing that rafting link, Erin! Definitely added it to my list.

I see Ospedale degli Innocenti on the list; on that same piazza (Piazza della SS Annunziata) is a church of the same name, Basilica Santissima Annunziata, with wonderful High Renaissance frescoes by some of that era's masters in the entry cloister; well worth a stop by. Entry is free.

https://www.visitflorence.com/florence-churches/santissima-annunziata.html
http://www.museumsinflorence.com/musei/Santissima-Annunziata.html
https://www.friendsofflorence.org/projects/cloister-of-the-vows-santissima-annunziata/

And while you're walking about admiring Brunelleschi's architectural designs in this piazza, look down southern end leading to Via dei Servi for a different view of his most famous accomplishment. :O)

https://www.istockphoto.com/photo/piazza-della-santissima-annunziata-florence-gm855153638-140635075

This piazza is very close to San Marco so if it's on anyone's list (Nancy, you've likely been there) review the open days/hours on that one so you can combine visits to both.

https://www.b-ticket.com/b-ticket/uffizi/default_eng.aspx.html

Check out the little wine windows...they are left over from the plague and have been revived because of Covid. https://www.cbc.ca/news/world/florence-tradition-wine-windows-covid-1.5887677

Grazie to all who contributed here. These are all such wonderful suggestions. You have helped me map out my upcoming 4 day visit to Florence!

Also notice the street art, especially Clet Abraham's altered street signs: https://www.cbsnews.com/news/trafficking-in-art-clet-abraham-street-signs-in-florence/

There was a small museum in Florence I visited 10 years ago and was so lovely and unexpected..."Museum of Precious Stones" I believe. All the pictures were mosaics and just dazziling detail....lots of semi precious stones used...