What to do with a hat

David this is what your day pack is for.

Maybe bring some kind of little clip to attach it to your belt. I've seen them but don't know where you get them. Someone else here will probably know.

In order to clip the hat on your belt loop or day bag the hat will need a loop of some kind. If there is a way to sew a loop onto the tag or somewhere inside the hat to the hat band that could work . The loop could be some kind of cord like from a shoe lace or such and any length loop that seems to work. Then use a carbiner (an S shape one would be good for this). Experiment and you will find a solution.

If you switch to a crusher style hat you can crush and stuff it into your pack or even a hip pocket.

Not sure you'd need a loop to use a clip, if the hat has a brim you might be able to just clip it to the brim. You might check an outdoor store like REI because I'm sure backpackers use clips like I'm thinking of.

This is what I was thinking of:
http://www.amazon.com/Elastic-Scarf-Clips-Attacher-Extra/dp/B00DV55YRA/ref=sr_1_2?ie=UTF8&qid=1427129473&sr=8-2&keywords=mitten+clips

I've been wearing hats and other head gear for about seventy years and I hold my hat or cap in my hand. If I have one that rolls up I put it in my pocket.

There are companies that sell roll up hats, quite a few, matter of fact. Some are felt, some are fabric, some are panama. A perusal of the internet will help you. David Morgan and panamahatmall are a couple I'm familiar with as is Islay Woollen Mill.

Carry it in your hand. We've been working on that convenience for several million years.

I thought the OP had just bought a special hat for his trip and it doesn't sound like it's a 'crusher' type of hat, maybe I'm wrong. And I think he was talking about if he had to remove it and his hands were full of other stuff.

I'm with Monte, I wear hats all the time and never have an issue holding it while inside. You learn to hold it and your camera at the same time. And really the only time I don't wear it is in churches for respect or while sitting in restaurants. I don't take it off in museums usually.

In many museums in Italy it will not be a problem, because you will not be holding a camera. Not allowed.

Lola, many museums have relaxed the "no foto" policy but flash is still not permitted.

True: they did away with the no-photo rule at all Italian State Museums last year (but no flash or tripods).
Private museums and individual churches may have different rules, though.

My mistake. I suppose it became impossible to enforce the ban once people started using their phones for photographs.

So the Villa Borghese, Sistine Chapel, and Museo Correr in Venice are the exception in still banning photos?