The amount one can stretch the opening of a hat will depend on the quality of the hat, a wool felt is best just to reshape the opening longer or wider, but a fur hat can often be stretched at least a size or two bigger so if you find that nice old 100% beaver at a garage sale for $5 and it is a a size or two small, it can be stretched to fit the head with some care.
There are different kinds of hat stretchers, the one in the first picture is the one I use a work, I don't know how old it is, but it at one time had a cord to run a heating element, the cord is cut off and it has cloth insulation on it.
The second one is the kind most folks will own, this one is wood, but most are now plastic, they will do a good job and sell for $20 dollars or so. So get your tea kettle hot and steaming, they work as well as the electiric ones we use at work or even get a spray bottle of water and go to the next post, we are going to stretch a hat.