Looking for suggestions on securing floor tiles
Nothing really bounds well to small pieces of loose thinset.  The problem is that the thinset that is adhered under your tile isn't in one stable piece.  Meaning that even if you add more thinset to a band of loose thinset it probably won't hold long, it certainly won't hold long term.  

I suggest you cut your loses.  You have had someone replace the tiles.  I hope they told you they wouldn't last.  For one I would have never just replaced your tiles.  It is just too obvious that there would be issues really soon, since you are doing the same thing that just broke.  I suggest demoing tile and doing it right.  By the time you try to do 4-5 rounds of touch up you will be at the same amount of time than just redoing the floor.  I mean there are instances where a plywood subfloor for tiles would conceivably work and may last years but you know that yours is not this case so why bother gluing the tiles down?  They will come up, more tiles will come up, you will have grout crack everywhere.  [And I am not a big demo and do it perfect to last 200 years Mike Holmes type]