It's a problem with all cameras at small f -stops. Your S6500 is limited to F11 by Fuji because of it.
I try to keep my S6000 between f4 and f5.6 if at all possible for maximum sharpness.
Canon PowerShot A570 IS
Cameras from my past:
Fujifilm S6000fd; Canon PowerShot S50; Canon PowerShot A40; Nikon N8008s; Pentax Spotmatic F; Fujica ST605; Canon Canonet GIII QL19