What happens when you increase the concentration of conjugate base in a buffer system?

Suppose that we add some NaOH to a weak acid (HA), and the pH of the buffer system is at 6.

HA + OH- → H2O + A-

If we add the salt of A-, does it change the pH (after the new equilibrium is established)?

I would guess that the pH is maintained because the ratio of [A-]/[HA] would remain the same, but then as reaction is shifted to the left, [OH-] would increase, which would mean higher pH at the new equilibrium..

​

I know that adding more A- to a simple aqueous solution of HA (without NaOH) would increase pH. Would there be a difference for a HA + NaOH solution?