This might work unless your first flight is delayed even a few minutes. You will then lose the second flight, and unless booked on the same PNR, the second airline is not obligated to put you on a later flight - it is as if you just missed a flight by arriving late to the airport.

Make sure to plan plenty of stopover time between flights to avoid this. I would not schedule only 1 hour between flights as I would be worried than any delay at all would leave me stuck. Then you have to buy a walk-up fare at astronomical prices or pray that UA would take pity on you and reschedule you on a later flight. I'd estimate that might happen about 25% of the time.

Otherwise, just get from one flight to the other; depending on the airport, you may or may not have to go through security again.