The change that would make finding the next solution maybe not quickly would be to the basis matrix and its inverse from the last solution. Still, finding a new basis inverse would be much faster than starting from just the whole problem. Yes, changes in the basis matrix could make that basis infeasible, but could attack that the usual way with artificial variables, and there may be still better ways.

Changes in the costs c, the right hand side b, or the non-basic columns of matrix A would let the new solution be found usually quickly.