If it's just a skin inflammation, a systemic antibiotic may not be necessary.  I try to avoid them until it becomes necessary. Try an antibiotic cream. I found that triple antibiotic ointments based on petroleum jelly seem to bother them. If you don't see improvement, that would give you some information to work with. The inflamed oil gland was obvious; swollen and red and tender. She was pulling her tail feathers out. I would suggest you also try the hen saddle (there are some instructions for making one on here somewhere), and see how that affects the area. One vet used a piece of tape to look for parasites or bacteria. He pressed it against the skin, pulled it off, and looked at it under a microscope. Vets here cost $65 for the visit. Treatments and medicine cost more.
 
To find mites, go into the coop in the dark (if it's accessible) with a flashlight and turn it on suddenly. If they are there, you will see them scurrying. Some you have to look at the base of the feathers to find. Mites are usually on everyone, though. There are poultry dusts with permethrin (or pyrethrum, not sure) which is a natural insecticide and repellent. I put it in a sock and lightly patted it over them while ruffling their feathers. Try to keep it from their face. 
 
I have found that bare area on the bottom to usually mean internal parasites, but for a recent one it was an egg laying problem. Is she laying eggs? Safeguard is a very safe medication for parasites. It's difficult to overdose, but you do have to avoid the eggs for 7 days (per European Union regulations). If you have different breeds laying different color eggs, it's easiest to treat one breed and avoid those eggs, then treat the others and use the eggs of the first batch. That does increase the risk of the first batch getting reinfected. 
 
Hope some of this is helpful.