You have to charge more as a contractor because you have to pay for your own insurance, office supplies or computers, etc. The hourly rate isn't an apples to oranges comparison if you are a salaried employee.

I think I'd just ask if they would be open to moving you into that new role permanently if you did a good job with a salary increase. If you could work from home as a contractor that would be something to consider, but doing contract work comes with a lot of other annoying shit like the aforementioned benefits and tax stuff, not to mention less job security. Plus I think it would just be kind of weird to tell your company that you're quitting but you want to be hired again as a contractor.