That's a really good thought. And make sure you let the clinic know that you are ready and willing to snatch up your cat and go at any moment.Unless you have a relationship with a clinic in your area, I am very doubtful that you will find a clinic that will accept payments. It's nothing against your personally (though I know it feels that way!) but most vets have had people take advantage of payment plans and bail before the vet collects anything. These people put others who DO intend to pay off a bill in a bad situation.
I would keep calling the low-cost clinic (every day or every other day...you want them to know you!) and ask if there have been any cancelations or sooner opening available. My guess is that they will get tired of your calls and do their absolute best to get you in sooner!