Part 2 of Watership Down takes place during June, since the last mention of "May" as a month comes in Chapter 14:
As they came up into the May morning...
After this, the first concrete mention of the month in Part 2 comes in Chapter 18, which is the first chapter of Part 2:
As Hazel still went up, the south wind began to blow and the June sunset reddened the sky to the zenith.
In Chapter 30, the first chapter of Part 3, the following quote occurs:
June was moving toward July and high summer. Hedgerows and verges were at their rankest and thickest.
(Adams, Watership Down, Google Books)
From these quotes, it can be seen that Part 2 of Watership Down takes place entirely during the month of June, before the hottest part of the summer but well after the heavy snow and rain of the winter has ended. This allows the rabbits to travel in relative comfort, with easy access to food plants and cover. However, it is also the time when many predators are at their most active, some regaining their food from hibernation, and many gathering food for new predator infants.