I think it's awesome that you want to ride more and leasing a horse is a great step to take if you're not ready to buy. My first horse was leased and I learned so much from him.
If I were you, I would talk to your trainer and see if she knows of any horses for lease. If she doesn't, talk with as many people at your barn as you can...there's a good chance that one of them can point you in the right direction.
Leasing is fairly simple, but you just have to be sure that you and the owner work all the money issues out: how much your monthly payment will be, who is going to pay for vet and farrier bills, feeding cost, etc.
A few more things to consider when talking with the owner: Will you be able to show the horse? Who is responsible if the horse gets seriously ill or injured? Are you going to be the only person using the horse or will you be sharing it?
Despite all the cautions that you will need to take, leasing is probably be the best option for you. Just be sure to do your homework and you'll get into a great situation!
Good luck! 