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It is not unusual to hear your signal on other frequencies when very close to the transmitter. This could be due to receiver overload or other artifacts which may not be an actual signal. But, since you hear it farther away it may indeed be a “spur” from the transmitter.
A few things can cause this. Without being able to put my hands on it all I can do is offer some suggestions. The first thing I would suspect is the power supply. A supply which is “soft” to RF could be a problem even though there is a good bit of bypass capacitance in the transmitter from power in to ground. I have seen problems such as this with VHF amplifiers, especially when used with a regulated supply. Try another power source.
It could also be that the loading coil/antenna is not actually resonant at your operating frequency an it is preferentially passing this artifact from the transmitter or providing another resonance for the output circuit. Try changing taps and see what happens.
Also, thanks for reporting on your loading coil success (despite this minor problem).