enhance instability of transistor for oscillator design

Any suggestions to enhance the instability of transistor over a wide range of frequency inorder to design a oscillator. Tried using a parallel LC on the source (with common source config), but unable to satisfy multiple freq points.

Using only one transistor as an oscillator, to get the widest frequency range have to increase as much as possible the open-loop gain of the stage, and to lower the Loaded-Q of the resonator (to the limit of starting the oscillation, and getting good phase noise compromise).
The widest frequency range can be achieved using a differential push-push oscillator type (two transistors, and one resonator).

I know this seems a bit obvious, but be sure the transistor still has gain to something like 2 or 5 times the highest frequency of operation.

It would help to know the frequency range you have in mind. The resonant structures that suit one band only mess up those for another. At HF, one has to switch component sets and gain-setting components and use several oscillators to seamlessly cover the range.

If by some chance the frequency range you require can be covered by a fast clocked DAC and pre-set sinewave accumulator, such as from a modern digital synthesizer chip. You need only a small low-pass filter after to remove the clock, and you can generate waves at will.

BUT.. you did say one transistor. Using the suggested 2-transistor push-pull method definitely has more potential, and was the basis for a HF gate-dip oscillator design that spanned several decades frequency. The resonator components and gains still had to be changed (using plug-ins) to cover the range ~1MHz through to 480MHz.

Wideband oscillators can be obtained by using low Q valued resonators with the expense of poor phase noise.
The active portion must satisfy oscillation conditions during whole frequency band ( negative impedance.. ) and varactor should satisfy wanted capacitance value in this band.
But don't forget, loaded active part by the resonator may be impacted much.