The road leaving Erandique  to the south heads up over the mountain behind town and continues on to Mapulaca near the Salvadoran border. Little traffic passes on this road even in the dry season, but hitchhiking is possible if you’re patient. It might be quicker to find a guide and walk by trails through the lovely countryside.
From Mapulaca, a dirt road winds its way north via Valladolid and Tomalá to La Labor, where it meets the Nueva Ocotepeque–Santa Rosa de Copán highway. This is serious adventure-travel country—pickup trucks and hiking are the only means of transport all the way. Don’t plan on getting anywhere quickly.
A couple of veteran Honduran travelers report that this is one of their favorite parts of the country, with friendly folk and spectacular scenery, as well as horrifically bad dirt roads. Decent hostels and basic food are available in both Tomalá and Mapulaca, and probably elsewhere too if you ask around.