Travel within USA

You will encounter 3 checks on an internal USA flight:

• TSA gun/bomb/ "you have a ticket"* check.
• Boarding pass check by gate agent.
• Uber driver at destination checking that you are you.

The several States which united to form the USA already agreed on freedom of movement and commerce, so they also agreed on a sort of "super Schengen area" among them. The states entirely defer immigration to the Federal government, and take no role in immigration. Since the USA is also a trade union, there are no customs controls either, except for very minimal health inspections, e.g. California will not allow some seeds, animals and foods, to stop infestations nd invasive species. (Some nitwit seeded a Sierra reservoir with Northern Pike; had that reservoir spilled before this was discovered, it would have wiped out California salmon.)

So, the facilities simply don’t exist to check and stamp your passport. There’s not a "line" for that in airport layouts.

You will enter via TSA security (gun/knife/bomb check) into the domestic "sterile area" (sterile of guns, knives and bombs) intended for departures, with the wide selection of overpriced restaurants and shops. When your flight lands, you will exit the plane into that exact same domestic sterile area intended for departures. Feel free to stay inside the sterile area for a bit and overpay at a restaurant or shop 🙂

Notice how that differs from an international arrival, where you are pushed into an austere area with little but immigration.

You exit the sterile area, typically toward baggage claim. It will be a hallway that has a TSA agent posted, but their only job is to make sure nobody tries to enter the sterile area that way. They will ignore you. Their job could be replaced by a mechanical gate/turnstile, but that would confuse travelers and create a jam-up.

Once you go down that hallway, that’s it. Welcome to New York, enjoy your stay.

* As such, they check your ID, but mainly to assure the ticket you have is yours. In the 80s, TSA didn’t care if you had a ticket. In exploring cities with good public transit to their airport, I would often enter the sterile area and enjoy some of the more exotic restaurants there. I hope we are able to return to that someday.

It’s sufficient to have state driver’s license or state ID card to travel inside US, which means that visas are not part of TSA job on internal flights.

I have been traveling within US while on expired F1 visa (still legally allowed to stay but visa expired). Not once have I been questioned about visa status on internal flights.

I believe the existing answer and comment are slightly confused by the terminology used in the question; there is a slightly idiosyncratic vocabulary in common use by H-1B workers in the US. Rather than try to correct this through comments, it seems more efficient to post an answer. Here is a translation of the question, with the modifications in boldface:

I am an H-1B visa holder. My visa is valid until September 2023. My visa extension is already approved and I have my hard copy of the approval. My question is can I travel within USA like I want to travel from Texas to New York for vacation?

I know that if I leave the US then in order to return I need to get a new visa in my passport, but to travel within USA in flight, do I need to follow same?

As noted in the existing answer, you do not. You do not need to maintain a valid visa while you are in the US.

You do, however, need to maintain valid status, as you seem to be aware. Since you’re flying from Texas, I have to mention the US Border Patrol internal checkpoints. Border Patrol can check your immigration status at these checkpoints, which are allowed within 100 miles of US borders, so they occur fairly regularly in southern Texas, including at McAllen and Brownsville airports. If you are flying from one of those airports, you should make a point of bringing the hard copy of your approval with you. You should also bring it whenever you travel by road to any point that is within 100 miles of the US-Mexico border.

If you’re flying from another airport, it’s unlikely that you would be asked about your immigration status, but you might want to bring a copy of the approval for peace of mind.

Internal flights within the US do not involve visa checks or passport stamping. Just bring a valid passport or US driver’s license to show TSA.