I think flying fits into two categories:
1. Flying something challenging and interesting - Tiger Moths, Yaks, Husky, Cub, Pitts, etc.
2. Fly a sensible tourer - TB20, DA40 and 42, Mooney etc.
The first group are always fun and a challenge.
The second group are not much of a challenge, nor that interesting for pottering around the local cabbage partch but far better at getting you from A to B at a reasonable speed and without too many concerns about the weather.
I reckon that most aircraft fit into one category with few that cross over.
If you are lucky there are some more specialist schools and clubs that offer aircraft in the first category. They arent on the whole used for ab initio training so availability is reasonable.
There are plenty of schools with aircraft in the second category (although not necessarily the examples given). However, fewer schools have tidy examples that they are willing to let you take away for any length of time.
Plenty of groups with aircraft in both categories.
Unless you have places to go regularly and given your experience and the type of flying you do, you might find something in the first category the answer combined with a bit of hiring when you want to go further a field.
If you prefer to fly something in the first group your options are better. Almost certainly you dont want to fly far so if you are lucky and there is an operator with aircraft of this type far better to rent because availability is likely to be good.