Widen a hole in an existing knife block?

There is only one possible way of doing it that I can think of without cutting the block apart on a table saw and re-gluing it. You would need a power tool called a "reciprocating saw", aka a "Sawzall" with a long wood cutting blade, long enough to get to the end of the block.

You would have to clamp the block into a woodworkers vise or clamp it down to a table, then using the longest blade you can get, run it in the slots multiple times to make them longer. The blade will likely be thinner than the slot, hence multiple passes, trying to exert pressure to one side, then the other to chew out a wider kerf. Don't expect it to look nice though, the edges will look chewed out.

Sawzalls are around $250 on line. You might find a cheaper one that will cost less, but likely only last for this one project. Tool rental yards offer them though, if you have one near you. You will have to buy the blade if you rent the saw though. The blade will cost you around $15, so this had better be important to you, compared to just buying a new block.