How oriented is the ferrite rod in the Sony Dream Machine ?
If the ferrite rod is from front to back (not from left to right), that could explain a good coupling with the spool of wire.
Then, the spool could have some sort of tuning by inter-capacitances.
Try this: get a card/plastic/wooden box about 10 x 16 inches, wind 18 or 19 turns of insulated wire around it (leaving about 1 wire diameter between the turns), connect the ends to a variable capacitor (any standard one for AM tuning, using only the widest gang). That should give you quite a boost ! (Small VC looking like a dice, from a discarded AM-MW radio should do the trick. Don't use bare wire because any contact between the turns will ruin the tuning). You need about (10+16)x2x20 = 1040 inches, about 85 ft, of wire. Think first about how you will affix the VC to the box, and how you will affix the first and last turns to the frame, and leave about 20 inches at both ends to connect the coil to the VC.