This may be a case where bias and collector current combine to sustain tank oscillations. The tank only needs a tiny boost of energy during each oscillation to replace lost energy.
The transistor bias turns on in a range from .3 to .7 V. So as you say that may set the upper limit you can get for voltage swing. Any greater V and it overcomes the zener threshold in the transistor. (I guess that's what you mean by triode mode.)
Are you using one transistor to sustain a tank circuit? I have not seen schematics for 1-transistor tank oscillators. I've seen Clapp, Colpitts, Hartley, twin T, phase shift, etc.
Sometimes you must let it be enough for one device to perform one purpose. Therefore it may be necessary to add a second stage (second transistor) to get voltage amplification.