If you are planning to go around now don't stretch out into may. May through mid October best to avoid. Unless you want to take advantage of low season
Mostly it's very hot but in October till February it's good n more you will see festival celebration at the time like pushkar festival Jaipur festival n many more good luck
I was in Jaipur for 4 years during my engineering days and visited Udaipur, Bikaner, Ajmer, Pushkar, Jaisalmer and even Jodhpur. So I think I am eligible to answer this query of yours.
If You love chilly nights then obviously WINTER will be the best season
BUT
If you're "neither too hot and nor too cold" kind of season loving person then SPRING /AUTUMN will be the best suitable season for touring whole Rajasthan. Good Luck! Happy Traveling!!
To be honest ; please wait till September; because I recently visited rajasthan and temperature there gets more than bearable range in march only. So, yeah September to March I think.