Tread very carefully through this minefield, it will depend heavily upon which type of aircraft you refer to and how it was certified.
V1, as you know, is the latest speed at which you can stop, but also the earliest speed at which you can go.
In the rejected takeoff case, most of the run up to V1 will be below Vmcg, and, if a stop becomes necessary the throttles / thrust levers are chopped rapidly (removing any future problems with directional control below Vmcg) and the aircraft stopped.
Beyond V1 (and most operators specify at or beyond V1), rejected takeoff is impossible and you are commited to 'GO'. This may involve considerable acceleration to Vr on the ground, but, if below Vmcg, such operation would be impossible due to insufficient directional control. This is why V1 must be at or above Vmcg (the margin above depends on the certifying authority).
The simple solution is to increase V1 to Vmcg (as you've alluded), and manuals for some aircraft I've flown such as B777 advise to do this. Here comes the tricky bit.........
V1 normally provides for either a safe stop, or a safe continued takeoff. If the V1 is arbitarily raised by a pilot, then his rejected takeoff capability within a restricted field length just went out the window - Can't have this No.1! If the pilot arbitrarily used a lower V1 than certified, and Vmcg was not a problem, rejected takeoff performance would be much improved, but the much longer (slow) acceleration from V1 to Vr and / or V2 would require a Takeoff Distance beyond that available - Can't have this No.2!
So what now? If the Approved Flight Manual advises to raise V1 to Vmcg, the manufacturer has factored this in to the complete calculation package to ensure that, with this pilot modification, both the rejected and the continued takeoff can be accomplished. DO go looking for performance penalties which may apply!
If V1 is below Vmcg, and the Approved Flight Manual makes NO mention of raising V1 to Vmcg, the aircraft is in a 'no-fly' zone and takeoff must not be attempted. If faced with this, your only solution would be to find a lower takeoff weight where V1=Vmcg, and use this as your Maximum Takeoff Weight. Now, both the rejected and the continued takeoff can be accomplished.
I hope that this clears the air.