Sure. Look into some complicated particles with different components.
You have to look exactly how to setup lifetime of particlecomponents, they have to be different in your case.
Make a moving object in your world, bind the particle onto it, to test result.
Ok the center of the thrust has to die quick, the longer flametrail has to live longer(longer lifetime = longer trail). the center of flame has to die quick, because you want to suggest a heatsource fixed at the rocket.
Setup two different components for testing, use firexample.
Or here, not to bad as starting point. 2.5 - Ribbon Trails Streaming from Other Particles | Unreal Engine Documentation
For the starting sequence, i would make another/second particlesystem, like an explosion.
Startsequence trail starts, explo scaled down to zero.
While small amount of time, scale up explo, while spacething lifts up, scale down again.
Or calculate distance and use that to scale.