This obviously is a big one - more so if your teams are based in the US - the extreme is the west coast of course. There is no one-step easy way to solve this: asking folks in India to work through the night to give your teams 100% overlap is not sustainable, even if you pay a lot; carving out mutually exclusive work for the teams so overlap is mitigated can work initially but can create a sense of work disparity over time.
The most important step towards solving for timezone difference is accepting it - this may sound simple but its been amazing for me to see how treating the timezone as a reality (as opposed to an obstacle) can help your teams adapt and make some changes as well - like having daily sprints at 9am EST instead of 2 pm EST so its easier for your colleagues in India to join. The faster you build trust and rapport, the easier it will become to adapt to the timezone differences.
If you have a relatively large team to build, the squad lead or site lead is a great idea as your first hire - this person not only serves as the translator of work and as bridge for building trust, but also buffers your local India team from too many late night calls.