I'm not sure what sort of FLEX you guys do, but with ours (and every other one I've seen) a reject AT V1 will (in theory) get you stopped before you run out of runway. That's the point of V1. If you couldn't stop in time you wouldn't be able to FLEX. "A few hundred feet"? Honestly now...
That's the definition of V1. We used a MIN V1 often with a flex but still, it is that speed to which you can accel and either takeoff or stop in the remaining runway. If you can't do that, you don't have a valid V1.
There are, as many have pointed out, many problems with high speed rejects. And something not pointed out in the discussion thus far is ANY reject is reported as an event by the tower.
But the heart of a reject. No two rejects are the same. And to conduct a true high speed reject without further problems is no easy task.
For those with autobrakes, studies show that pilots still want to get on the brakes which will disconnect the autobrakes.
For those without, the rule of thumb is if you can shove the handbook under your butt, you are not standing on the brakes hard enough.
Bring the airplane to a stop on the runway and assess the situation. Tower is going to be wanting you to get off the runway but the
runway is yours so don't be in a hurry to give it up. Once you have determined you have no brake fires, no further complications, decide if you can exit the runway and then what you will do.
Note too that most taxiways are more narrow than runways so if you get on a taxiway expecting emergency equipment to reach you, know they will be able to maneuver around you. The British Midlands (?) incident illustrates what can happen where the winds blew the fire under the airplane, the ground was soaked and the emerg equipment mired up in the grass trying to get to the airplane.
And there has been some problems with the commands to keep people on the ground or evacuated. Many, if not most airlines have gone with "REMAIN SEATED" or "EVACUATE". HuFacts found that if you used "Do not evacuate', the only thing people heard was EVACUATE. More than a few incidents have occurred where pilots looked up at the annunciators and saw doors opening.
The problem not often discussed is with a balanced field, when a situation occurs at near but just below V1, without almost instant response, you may find yourself in an unsolvable situation, not enough runway to get airborne and not enough runway to get stopped without using the overrun or clearway.
Finally, many operators can not agree on whether or not to advance thrust on the good engine if an engine fails. Some say leave the thrust as is because the data was based on an engine failure at V1 and advancing the thrust will only introduce more control problems (additional yaw). Others suggest going to max thrust if you had been using a FLEX or reduced.
The summary is.. know what the procedures are. IF the throttles come back, there is no changing of the minds. REJECT. NEVER advance them again hoping to get airborne. Likewise, a short while back, a Lear at Aspen (?) proved that if you land hot and extend the speedbrakes, do not try to stow them and go around.
Oh, one final point.. just remembered. STOP STRAIGHT AHEAD. The SW 737 at Burbank probably would have stopped if the crew had not tried to turn. The turn caused the tires to break traction and then it was over. I got this info directly from one involved in the investigation so it is valid.