There are two schools of thought to getting your fingers to cover the holes.

The Irish method - 'piper style' - don't use the fingertip pads, push your hand past the holes, and use the next joint on the fingers.

I play my large whistles in the 'flute style' - the hand is angled relative to the tube. One way to get there is to hold the whistle sideways (as a flute) then rotate the whistle back to normal playing position, sliding things around but keeping the holes covered. I end up using the sides of the pads of the tips on some fingers.

Whatever you are more comfortable with.

When attempting whistle surgery - work slowly, take care and think it through first!