Janitors will help with laundry and can enter any room regardless of the security coding so additional janitors should help.
Although they were pretty stable in the general population, I built a separate wing coded for supermax prisoners (Security Sectors menu) and put them all on permanent lockdown in their cells. I added a small laundry room nearby coded for staff and assigned it to the supermax block and death row (Logistics menu) so the janitors deliver clean uniforms there.
Alternatively if your max-sec inmates are free to leave their cells then you could build a small laundry in their wing and code it for the max-sec category too so the inmate workers assigned there will be able to access the cells. Ditto protective custody.