I won't say the place as offbeat but it's a definite treat.
Trek to Tadiandamol, the highest peak in Karnataka.
The best way is to start the trek early in the morning from the base. Have a full day reserved for this since the hike would probably be 10 Kms on foot. Carry proper trekking stuffs and water bottles. The view and the way is absolutely worth it.
Cheers.