First the “new illness” would need to be a previously undiscovered bacterial infection. If the “new illness” was not caused by bacteria then antibiotics would not work.
If we then have a few patients with this new disease. we would culture the bacteria and grow them up ad test all current antibiotics against them. It is likely that at least some of the antibiotics would work. Many bacteria (even undiscovered ones) have similar biology, this means that the antibiotics will often kill them all. If we couldn’t find an antibiotic which would kill the bacteria, then we would have to go back to the drawing board.
We would have to find out why our current antibiotics don’t kill it. We can then try to develop chemicals which allow current antibiotics to work or we could try to find new vital targets present in the bacteria but not in human cells. Once we have a target (either to stop resistance or to specifically kill the bacteria) then we would design chemicals to target a specific protein to stop the targeted process. If this works, then the chemcials will be optimised, tested, etc. eventually forming a new drug. Though this is a very long and expensive process and probably wouldn’t be worth it for one bacteria!