If you're talking about a crude 'bug' there's little you can do, but assuming it's a proper RF signal generator use VERY high quality components, make the coil VERY thick and strong, and build it inside a temperature stabilised totally screened metal box.
perhaps, even if you are given ways to stablize, you may not have that level of Lab quality test instruments that could measure the stability aspect. In principle, we cant stabilize any oscillator just by normal means.
the best for you would be to go for a standard reference oscillators that are sold . they are periodically calibrated with rubidium standard clocks. Of course they cost a moon.
the normal means of stabilization would definitely distort one or the other parameters.
we can't just have calibrated quality, voltage, resistor, capacitor, inductor , nothing. it is a profession by itself and costs.
that transmitter you posted, is it good? it looks real simple, almost too good to be true.
And if it is, can i switch the transistor with a bigger one and up the voltage so that i get more power?