Even though Trevelin is a small town there are many schools, including English institutes. Unfortunately the province of Chubut has had many problems of government administration for many years and this makes strikes quite common. There are private schools, but they are much fewer and not especially better than the public ones. Remember that public schools are quite good in Argentina (in spite of what “people say”), outside the big cities they are the only option and the best teachers, the ones with the best vocation work there.
If the options in Trevelin are not convincing or there are no vacancies available, there is the alternative of Esquel, which is 25 km from Trevelin. It is not that far, but it is 30′ by car and on some winter days there is ice on the road which makes it impassable.
The choice of school is something extremely personal, the reality is that the options are very varied so the recommendation is to go to the page of the ministry of education of Chubut that has a beautiful interactive map, lots of info about everything they have under their coordination and then you have to come and choose. If you go on an exploratory trip around October or November you will be able to talk to many principals and get to know the schools, they are all very open to give you a meeting from one day to the next.