Don't know of anything around Teguise. If you are hiring a car then Papagayo isn't far away.

Do you mean Teguise, inland, which has a large Sunday market or Costa Teguise?

Sorry, Costa Teguise.

We'd certainly recommend Charco del Palo, an unspoilt naturist village.
Go for a short drive N on LZ-1 and shortly after Cactus Garden get into the next village and turn right at the Tapas Bar, just after the bus stop and second sign for hospital.  The narrow road winds down through cactus fields to Charco del Palo and park anywhere.  If you haven't already stripped, do so, leaving your clothes in the car and wander around and along to the coast and enjoy cliff-top walks in either direction and dips in the various tidal pools.  You can then assess what you feel about the place for possibly booking your next stay there.

I wholeheartedly agree with Listerboy re Church Del Palo which is virtually on your doorstep.

The cliff walks in each direction are superb as is the sense of natural exhileration. Well worth more than one visit and a future stay in the village which has a cool laid back feeling from end to end strictly not for the party people.