When you say "In a single stretch of flight," do you mean flying without landing? Or flying from one place to another? Sometimes birds, especially inexperienced birds or birds traveling long distances, become exhausted and have to touch down for a little. When they're rested up they get back in the sky and keep heading home.
How many miles pigeons can fly depends in part on the bird's physical build. Some birds are built for short distances, not long distances. (It's easy to tell which birds are which just by looking at them.) Short distance birds can still fly long distances, they just wont perform as well. Vice versa for long distance birds.
Pigeon clubs usually fly races from 100 up to 600 miles. Races up to 200 miles are considered short.
Heads up: the following numbers came from well-trained birds flying the distance that they're comfortable with.
In the club I was in, the longest races we flew were 500 miles. The fastest bird from our club could fly the 500 in 22 hours (ish). Under normal weather conditions, 500 miles shouldn't take longer than a day. Races of 700 miles have been finished on the same day of release under ideal weather conditions and with a tailwind.
The longest recorded official pigeon race was 1,100 miles.
I googled longest recorded pigeon flight (not only including races), and I got a couple of different results:
The longest homing pigeon flight ever recorded was 7,200 miles, from Arras, France, to Saigon, Vietnam. The flight took 24 days.
The longest recorded flight in the 19th century covered 7000 miles between Africa and England and took 55 days.
If you want to know how long flights under 500 miles take, I can tell you that, too.
I hope that helps! If I forgot something or you have more questions, let me know!