Hiking recommendations coming and leaving from Milan (6 days)

Hi all!

I will have an extra 7 nights in Italy after 18 nights in Central Italy with my Girlfriend. I have decided to leave from Milan instead of Rome and dedicate my last 6 days (7 nights) to hiking in the North from 18th to 25th of June.

From what I am gathering about the Dolomites, I may have waited to book accommodation too late and it is surprisingly expensive. However, I am also very interested in the regions bordering France and Switzerland which don't seem to get a lot of online discussion. Ideally, I would base in a local town or two do day hikes but I am open to staying in refrugios if it makes more sense on a route. My fitness level is decent but I am not looking for especially technical climbs where I would need crampons etc.

Does anyone have any recommendations? It looks like there are so many gorgeous locations North of Turin and Milan that just aren't getting as much discussion.

If it's relevant, I'm an Aussie with an EU passport so can look into hiring a car if really needed.