Hiking footwear suggestions?

Some will tell you that you need ankle support of boots, but as an avid hiker who cannot wear boots (they hurt my ankle bone), I am here to say that this is foot-dependent! If you mean this September, your time to properly break in a new pair of shoes/boots is fleeting, so I would suggest going to an outdoor outfitter asap to try some out.
I hike in low-top hiking shoes but they are sneaker type (Merrell, men's because the selection of women's shoes in wide sucks, and in spring and fall I go with goretex). Wearing properly broken-in shoes is the most important factor for you to consider right now.
If you decide you want boots or even just high top sneaker type shoes, you can wear those on the plane so they don't take your bag space.

ETA I think your sneakers will probably be fine, but you want them to be very sturdy (thick soles and not a lot of lightweight "mesh" stuff on the upper) because you want them to hold up against a lot of rock.

Also--don't discount the power of good socks, like Smartwool--they really do make a a difference!

When we did this tour about 10 years ago, we wore hiking boots. Whether you choose boots with ankle support, or low cut ones, I recommend a sturdy shoe that has good tread on them, and that are waterproof. Fortunately we did not have rain on the days we hiked, but we did practically every other day. I even find that I wear my boots in towns with cobblestones many times as they are so comfortable. I always wear my hiking boots on the plane; you can find some very light-weight boots these days. Currently I wear a La Sportiva boot that I got at REI sometime last year; this looks like the one I love, but might be slightly different -- https://www.rei.com/product/235503/la-sportiva-tx-hike-mid-gtx-hiking-boots-womens No breaking in required.

I swear by my Altra Lone Peak trail runners. I hiked all over Southern Utah red rock country, including across slickrock, in them. The zero drop takes a minute to adjust to, and they are runners rather than boots so I wouldn't wear them backpacking, but they are super comfortable and I feel very stable in them even though my balance is not great. I am on my second pair and never had a break-in period with either pair; they were comfy right out of the box. They are light and about as bulky as regular running sneakers. Mine are coming with me on my six-week trip across Europe this fall.

Go tonight to REI and try on lots of brands of waterproof hiking shoes. You need the traction and waterproof feature. Trails are well maintained, but are often loose gravel. My teens wore tennis shoes and regretted it on some trails. I love Merrells and Altras, but find what works for your feet.

Start wearing them immediately to get them broken in.

I'm a hiker, but not really advanced.
My impression from the Switzerland tour is that the trails there are rougher than trails in the US. I would want something more supportive than sneakers.

Trails are often muddy, so waterproof shoes are a must. My favourite for both every day walking and hiking are trail running shoes. Since these shoes are my every day shoe I mostly look for one that is not to flashy, just plain black or grey.

I have not done that tour, but I have been to most of the places it goes to. My husband and I are pretty big hikers, but nothing crazy. We have never taken hiking boots. We both wear regular Hokas and have never had a problem.

I've opted for hiking in relatively inexpensive running shoes ever since college. Back then, after it taking days to break-in my boots with the then state of the art hiking boots, I ran across a book which I remember being called the Complete Hiker (if I recall the name correctly) where the author described after a bout with blisters being forced into sneakers. After days hiking with sneakers, he never went back to boots.

Ever since reading that and thinking about it, I've done the same. A few times over the years I'd try new gore-tex thingies and trail runners but I'd always go back to just running shoes. Two pair strategy. Lightweight, quick drying. I've done extensive hiking on the Appalachian trail and the rockies and walked the camino more recently.

A month ago we were on the Rick Steves Swiss tour where they were some hikes in the alps. Running shoes for me.

Sure they get wet when it rains (as do hiking boots and trail runners) and they not as good as boots for muddy trails but the lighter weight and quick drying were the trade-off I've always opted for.

Others strongly disagree. Anyway, that's what I've done.

Happy travels.

You can get trail shoes now that are between sneakers and boots, and would be a good choice for such endeavors.

Merrell as noted above is considered the go-to choice and they are often on sale on the intergoogles (and have stores at outlet malls).

Speaking as someone who has done a lot of hiking in the Santa Rosa Mountains above Palm Desert CA, including very rugged rock trails, I've never understood the reason to wear anything heavier than good trail running shoes. Brooks, Saucony and Hoka will all stand up to that kind of hiking, and at trail's end the light weight will have paid off. (I wouldn't really trust Nike or Adidas myself in those conditions, but our hiking friends had no trouble with those labels.)

I second the thought about good socks. Smartwool are good, but I actually prefer the Darn Tough brand because they're a bit easier for my arthritic fingers to stretch onto my feet. But both will give excellent service.

When hiking I alternate between Hoka Clifton 9 (sturdy running/walking shoes with a good tread) and Salomon X GORE-TEX Low hiking shoes. I use Hoka shoes for dry warm conditions and Salomon for wet, cold, longer, or more difficult terrain conditions.

I used the Salomon shoes for hiking in the Swiss Alps in September/October. The trails can be muddy in places, and it may rain. They offered plenty of support.

A good way to handle wet conditions is wearing good socks and carrying a spare pair in a Ziplock bag.

Good solid water-resistant trail running shoes such as Brooks, Saucony and Hoka should work well, especially if you carry spare socks while hiking.

All the best,

La Sportiva Bushido III trail shoes from REI have served me well on various surfaces. The great thing about REI is that if you are a member of their Co-op you can return anything for up to a year with no questions asked. That includes shoes that have been worn.