Seems to me that any kind of hanging will work - so so many examples in movies and TV - where the character gets creative (if the captors take away the obvious belt)
- tearing up the sheets and creating a noose. (or their pants)
- a woman wearing nylon stockings used them to hang/choke herself.
OK -if no clothes...well, if the character is violently suicidal, they can always try to bash their head into the wall, sink etc.
Or - the character incites the guards (or fellow prisoners) to the point where the character gets killed.
One TV show ("House MD") had an interesting scenario where a death row character got himself placed in a temporary "solitary" holding cell (which was an office storeroom- copy paper and supplies) - and he ended up drinking the liquid copy toner-fluid.
Sharpening the toothbrush edge (against the concrete wall) could sharpen it enough so the character can slice his wrists.
If the sink/toilet is made of porcelain, then the character can smash it so there are sharp pieces, and slit his wrists.
Generally however, the method needs to be a quiet one (otherwise the guards will find him and get medical attention).
But if you are writing ... then you have creative license to introduce some interesting scenarios: The character stealing an implement from a guard or lawyer. Character making a deal with another inmate beforehand.
Choking with food? very much a stretch. However, you could subtly introduce a backstory where, for instance, the inmate was very allergic to some common ingredient (like pepper, or milk) ... but instead of telling anyone, he requests a meal with the ingredients.