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You would be traveling from USA to Hyderabad. As per the latest guidelines for inbound international travelers :
For all international travelers except travelers coming through flights originating from United Kingdom, Europe and Middle East
- Submit self-declaration form on the online Air Suvidha portal (www.newdelhiairport.in) before the scheduled travel
- Upload a negative COVID-19 RT-PCR report. This test should have been conducted within 72 hrs. prior to undertaking the journey.
- You would need to self-monitor your health for 14 days. There is no restriction on you traveling to any other place inside India.
Once you arrive in Hyderabad (Telangana state), You would be traveling to Chennai (Tamil Nadu state). None of these states have any mandatory negative Covid-test policy for inter-state travel between the two states.
However Tamil Nadu requires you to
- Generate an E-Pass for traveling to the state. E-Pass can be generated from here.
- 14 days of Home Quarantine for all passengers arriving into Tamil Nadu from other states/UT.
- Business travelers visiting Tamil Nadu for a short stay of 72 hours are exempt from home quarantine norms. In such cases the individual shall furnish valid return document
So if you want to travel back to Hyderabad from Chennai, you must do it within 72 hours to avoid home quarantine.
Telangana doesn't have any home quarantine requirements.
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