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Best fit: October to February
Cooler weather makes sightseeing in Bangalore, Mysore, Belur, Halebidu and especially Hampi much more comfortable. These months suit the itinerary’s outdoor walking and heritage-heavy days particularly well.
Also possible: March and September
March is generally dry and workable, but Hampi starts becoming considerably warmer. September can be green and pleasant after the rains, although occasional showers may still affect outdoor sightseeing.
Avoid / least suitable: April to August
April and May can be very hot around Hospet and Hampi, making long hours among exposed ruins tiring. June to August brings the monsoon; rain can affect walking, road travel and outdoor sightseeing, although the tour can still operate. Karnataka Tourism notes June–September as the state's main rainy period.