I'm aware that this question is very dated, but I wanted to comment since this has become such a hot topic once again in our area.
I can't tell you how many calls & hits a week on my website are from family's looking for a rental home. Rentals in Enid are limited for three main reasons: The oil business, the Air Force & the college (NOC)... oil business (at this time) being the largest. The good news is that it is not uncommon to be able to find a nice home to purchase in our area at a decent price (often less than what the cost to rent is in other parts of the country).
I'll reference my Renter's page on my website where I do list what resources are available for renters in this area. I also have information on that page for those renters who have a goal of home ownership down the line & what to do now to prepare for that next step.
It is a very tight rental market for upscale housing I have found. I would try" Rental" on TRULIA
also sometimes Craigs List.
The Enid News and Eagle paper has some on the weekend but they are usually gone in a week.
It really tough to find them.
More than likely you'll need to find a property manager to find one, but also ask around in the squadron or check with Tracy House in the Housing Management Office. There are several IPs that have rented to other military members after they PCS'd and that's the first place they advertise.
Here in Enid we have been extremely fortunate to not have most of the issues the rest of the country is having. Our job market is strong and our unemployment rate is down. We have had quite a few companies expand their employee base and they are moving here from all over. That is making rentals very hard to come by.