Please review: Packing List for Southern Italy in mid-May

Just keep checking the ten-day forecast. I have always traveled in late May and the vast majority of times, it has been lovely--warn days, cool nights. Last year was an awful deluge (in the north, not sure about how Puglia fared), but it was unusual.
I would take a very good raincoat unless your last forecast check looks completely clear.
For your cardigan/jacket search, do you have an REI? I find great stuff there, sporty and sometimes elegant enough for dinner. Their website has great sales.
Are you staying somewhere with a heated pool? I would not count on sea swimming.
I do think you may need a jacket for morning and evening.

Sunshirt? I used the Eddie Bauer Departure 2.0 sunshirt for my last trip to Italy and it was a workhorse. It was white but did not get dirty (and I am a slob!).

Sunscreen? I have very few that I can wear on my face without allergy so I have to take mine with me.

On the other hand, I have been in Calabria (admittedly not Puglia) at the beginning of May, and the weather at the coast was mild to pleasantly warm, but really cold at higher elevations, particularly in the wild area in the central mountains of Calabria. I think we even had some snow. Of course you aren’t going to be in that area, but I would check the elevations of your hikes and make sure you have enough clothes for cold morning and evenings.

in mid May, I would want something like this at the very least:
https://ibex.com/products/womens-shak-jacket-sale?variant=42684921512107&currency=USD&utm_medium=product_sync&utm_source=google&utm_content=sag_organic&utm_campaign=sag_organic&utm_source=google-ads&utm_campaign=&utm_agid=&utm_term=&creative=&device=c&placement=&gad_source=1&gclid=EAIaIQobChMIsYXC4ZXChAMV5l9HAR0BFg1CEAQYASABEgLmL_D_BwE
I like wool versus synthetics because it wicks moisture and resists odor.
Or, if you have a raincoat that could pull double duty, that would work.
You can always buy something there, but I really think more than a light sweater will be needed with nightime temps in the 50s--but I am pretty cold natured.

We were in Sicily in May and had unseasonably cool and rainy temperatures. I would not have survived without my rain coat (as opposed to rain poncho). I also often wore a very light weight denim jacket either alone or under my rain coat. For May, I definitely would want at least a sweater or something for cool temperatures that could be layered.

Thank you all for your rapid replies! As always some additions and clarifications. Last year we were in Northern Italy (Lakes Orta & Maggiore) in early May and there was a lot of rain and it was cooler than we expected (highs in the 60s) and I did wish I had brought another very light sweater (I had a cashmere poncho, but I just don't like that look, though I wore it nearly every evening :)). But hiking was totally fine in pants and short sleeves with the zip-up hoodie in the morning.

• Definitely will be checking the weather forecast as we get closer and will adjust accordingly.
• I do have a lightweight sunshirt that I will be bringing
• My rain poncho is a specialized cape for cycle commuting/regular use, not the type you pick up when you're somewhere and it starts pouring and you buy whatever you find at the drugstore :)
• The high temps here in May are usually in the 50s/60s, so the lows will not be that cold for us. For example it was 55 yesterday and I went walking with tights, a merino t-shirt, and my lightweight fleece/hoodie zip up.
• We do hope to jump in the water, but we're used to ice melt lakes and Pacific Ocean water temps. It just depends on the air temp that day :)

I really appreciate the input! I do agree that layering is the key and also appreciate that it doesn't seem like I'm missing something major. Plus there ARE stores where we're going!

Yeah with that light hoodie I think you'll be okay--it sounded like you were on the fence about it. They freeze me out on the plane so I never travel without a jacket of some sort.

Puglia is gorgeous, enjoy!

I'm curious as to how you plan to carry your personal item backpack through the airport/city, when you're already wearing a larger backpack. Does the Daylite attach to the larger pack? If so, does the double backpack make moving through crowds awkward?

Asking because we used wheeled carry-on luggage on our last trip since we needed day packs for our 8 day hike. I might change up my day pack for our next hiking trip if I can find one that will attach to my 7 year old Osprey Farpoint without being too bulky.

The one item I always bring from home in the zuS is sunscreen as last time I checked it was still way more expensive in Europe.
We spent ten days based in Polignano a Mare, loved it all!

I'm curious as to how you plan to carry your personal item backpack through the airport/city, when you're already wearing a larger backpack. Does the Daylite attach to the larger pack? If so, does the double backpack make moving through crowds awkward?

Carrying it by hand with the top grab bar is the plan. I did go back/forth/sideways around this and that plan may be terrible in execution, but that's where I am. If that gets too much I'll just wear it backwards on my chest, though yes it will look silly :). When I traveled fresh out of college 25 years ago that's what I did (with a bigger backpack even). The carry-on size for WestJet is one of the smallest out there for major carriers and the majority of roller bags on the market are too big, so the travel backpack was the best option for the 'big' bag.

If we weren't doing the walking/hiking I'd use a cute Sherpani or other tote/crossbody for my personal bag. But I'll need an actual sturdy daypack for most of our trip, so there we go. The Daylight is relatively small compared to most campus bookbags, so I'm hopeful it'll all work!

If you have access to a daily shower, that’s a lot of pairs of undies and socks to bring. I wash them when I shower, squeeze dry, roll in towel, then hang overnight and they are dry the next morning. In theory you could make do only packing 2 each but I always bring 4 each total , including what I’m traveling in.
I also agree an added sweater or hoodie type layer would be good in case it’s colder than expected. But I run cold and you may be one of those always hot people.
Have a great trip, sounds wonderful.

Your packing list looks great. Instead of a jacket you might consider a merino base layer. You can layer it under a long sleeve shirt and/or a windbreaker for warmth.