Check out self-adhesive Command hooks.

Looks like that is next to a door way, I would not use a towel holder there. You will be just looking for people to catch things on it a tear it off.

thanks all ill have a look at the command hooks and over door hooks. i was initially not wanting over door hooks incase the damaged the paint bug will look again.

I hang mine on the oven door handle.

I have an inside Command hook for my oven mitts and use the oven handle for the dish towel.

Definitely try Command hooks.

Everything depends on how you hang your towels.
All of my tea towel and bath towels have hanging loops sewn on,either at a corner or ,for longer pieces ,a loop halfway down the long side.
Bathtowels hang from sauna shelf pegs over the heating vent and stay warm and dry.
In the kitchen I have a lower narrow cabinet with a slide out steel rail over a heat vent ...room for four towels and sink pads to dry.
I also have the same configuration (leading to the laundry room,)as shown in your photo...I have screwed small brass hooks into the underside of the counter overhang,and have seven fresh teatowels handy there,no catching protrusions You may have guessed that I harbor a huge extensive collection of bath towels and tea towels also bed sheets ,shower curtains, window curtains and bolts of vintage
fabrics.It is a passion (or disease) but everything is neatly stored till the urge to sew a different case or chair cover occurs.