How many new volumes would each new cell contain whenever a cell splits?

The specific volume will be different for various kinds of cells. The safe answer would be that the new cell will pretty much have the same volume as the one that it divided from. This is true for most eukaryotic cells unless other factors like epigenetics or mutations come into place.

One example of moments a cell would increase in volume is during hypertrophy. This simply means that the cell is increasing in size (compared to: hyperplasia -- which is an increase in number of the cells). Hypertrophy is definitely an increase in volume of the cell but this doesn't necessarily translate to cell division (i.e. just because the cell is big now, doesn't mean it will still be big when it divides).

Another moment of increasing volume of the cell and now also related to cell division would be during the two stages in the cell cycle (i.e., G1 and G2 phases). This is the growth phase of the cell preparing to divide. However when mitosis or division happens, the cells will normally end with the same volume as when it started.

This are safe generalizations referring to the human cells. It would help if a more specific kind of cell was given.