Broadly speaking, more signed messages can get you more points on the curve you're trying to guess.

https://www.cs.umd.edu/~amchilds/teaching/w08/l03.pdf

May help if you're actually interested.

Edit: More signed transactions help with the classical and not the quantum part of schor.

Edit2: Schor has not yet even been able to factor the integer 35 with current quantum hardware, too much interference.