There’s no real concept of “faster” exhaust compared to yourself. The speed of the exhaust is relative to the frame of reference of your rocket which is an accelerating frame, that speed is always the same when measured against the stationary rocket in that frame. When measured from a fixed inertial frame the speed changes as your rocket accelerates.

For a more concrete example, for a moon rocket the speed of the rocket exhaust is around 3 or 4 km/s. On the ground the speed of the rocket is obviously 0, in low earth orbit the speed of the rocket is 7 or 8 km/s, and to initiate the transfer orbit to the moon you have to accelerate to about 10 km/s. (these would all be in earth centered, nonrotating frame speed measurements)

The rocket exhaust doesn’t have to get faster to get you to those higher speeds because you’re taking it with you.

The more you can increase the rocket exhaust relative to yourself though, the more efficient your rocket is.

Say a ship in 1D is flying at 50km/s, exhaust velocity is -25km/s, relative to a static frame.

Last bit of departure burn or correction burns will fly at 25km/s towards the ship, so if the ship decelerated to below that, the plume could catch up. Meanwhile, plume from deceleration continues at 75km/s away from the ship.

I was thinking that high Isp engines generally have insane exhaust velocity, like hundreds of km/s or more, that problems like this is not an issue even for interplanetary transfers. But interstellar is a bit different, depending on other factors such as dispersion, I guess?

A one dimensional rocket traveling from A to B: all of the exhaust emmitted after the rocket speed exceeds the exhaust nozzle speed will end up hitting B.

In three dimensions though in a hard vacuum, particles coming from a fluid with a bulk velocity of kilometers per second are clearly not going to be nearby the path of the rocket for very long at all