To solve the above problems with the rectangle coil (from a small transformer) I tried with some ferrite pieces from my coil collection. I had a separate coil from a disassembled block and simulated the groundplate with a piece of iron and the tuning block bottom with a piece of polystyrol:
Moving the ferrite into the right position and fix it with some pieces of a wooden match or the like and glue it – ready!
Best,
G.