Before I married my current husband,I had a duplex I rented with my 2 small children. I had no extra money and little space. I was renting and could not nail anything. I bought several square concrete blocks. They were like 12x12 and 4" thick. I laid 2 down and placed a 1"x10" board on top of it to make my own shelves. I made 3 shelves in a coat closet. The top shelf held my portable sewing machine. This closet was only about 3 ft wide and not very deep. Maybe something like that would work. It is solid but temperary. Back then the blocks were about $1 each not sure about now that was over 40 yrs ago.

Good luck and get well soon.

You asked "Suggestions for how to add storage shelves to a utility room coat closet without major cash outlay?"
I bought a set of put together shelves like the one in the link below and put it into a hall closet, works great. These type shelves just slip together with no tools needed and come apart just as easily. They also come in different sizes and I think sterilite makes some as well. They are also available at different stores....Amazon carries a wide variety of storage shelves too.
Hope this helps and you get better soon.

https://www.amazon.com/s?k=storage+s...f=nb_sb_noss_1

https://www.walmart.com/ip/Plano-32-...-Taupe/8251818

I have done the same... but an 18" deep one just fits in my large storage closet in my small apartment. The large clear tubs fit nicely on the shelves and i can see what's in them.

Such wonderful suggestions and great pictures...best of luck as you move through this hopefully temporary situation.

Constrained by limited help and investment, consider moving toys to the top, and quilting to the bottom. If the kids are visiting, the adult that brought them can reach the toys. Move towels and such to other storage, use some of your stash to make a basket for them. If you don't already have them, buy a couple more sweater boxes for quilt stuff. And one big plastic tub (or laundry basket) to hold bulk fabric on the floor. A small investment in two over the door hooks (or a really heavy coat hanger) will let you make a full door sized organizer with pockets for all the tools and such. And don't be afraid to call a local quilting friend to stop by occasionally to help with a fetch and store. Once the tools and basics live downstairs, it is easier to send someone else on reconnaissance to get project materials.