I hold an LL.B (3 years). where should I do a master's to enhance my chances of becoming an assistant prof at state/ central gov institutions?

Masters, NET & PhD is the usual route. Take a masters in something that you want to further research in later on during your PhD. Clear the NET, start teaching and when you have enough income go for a scholarship for PhD as it's needed to promote from asst to associate.

Mostly accurate with one modification. The PhD is now necessary even to become Assistant Professor Selection Grade (grade 3), a post that an Assistant Professor gets by promotion after 10 years of work.

Thanks for the update. From where can I see this hierarchy of promotions ? Any links ?

Just search for UGC CAS Guidelines latest version.

Thanks 👍🏽