Cell phone service recommendations for Ecuador.

Our cell phone is MovieTime.  They have the best rate on prepaid phones.  Cell phone contracts here are 18 months.  We will not be here that long.  We also have TV, phone and Internet with Claro.  Good deal compared to what I paid in the US

You probably mean Movistar... ok, you only sign an 18 month contract if you, let´s say are getting a new phone from them with a discount. If you bring your own phone, there is no contract, and you still can get a post-pay plan, which is more than that $6 card you load every month, but cheaper and a lot more convenient if you use your phone more than bare minimum. I, personally, am a fan of Alegro (it is CNT now). Cheapest rates to US too.

We talked to several companies and none of them would provide us a plan on our own phone. They wanted an 18 month contract to get service with our without a new phone.  We purchase an ATT Go phone in Wal-Mart for $11 before we left the US.  It cost us $10 to get it modified and another $5 for the MovieTime SIMM chip.  We buy minutes at the drug store and get 3 for the price of 1.  I put $15 on the phone, got $45 of time and have been using it for months.  If you can convince one of these providers to put service on your phone without a contract, that is great.  We tried all of them we could locate downtown and none of them would provide service without the 18 month contract. It could just be the people selling the service. They probably don't want to open that no contract door or everyone will be using it.

You really surprised me, Fred. Probably, it is just the people. I brought my blackberry and got a post pay (billed) service on it with BB internet with CNT (Alegro back then), $10.00 + IVA + $14.99 (BB Internet). This is sort of credit from the company, so they do ask for all Ecuadorian papers, like cedula, electric bills etc, but they asked if you want pre-paid or post-paid service, and I suppose if you appear on all their credit systems as clean, should be no problem.
P.S. "several" companies... there are only 3 companies. Did you go to the central offices of the respective companies and not to the small cell phone store? Coz, those small stores just want to sell you pre-paid, nothing else.