One of the tried and true axioms of your golden Age SF and most settings is that there a single way to travel faster than light. FTL itself is fairly common in SF. Even hard SF manages to get away with this one cheat. After all you're breaking the rules of physics as we know them once (maybe twice if you have anti-grav), there's no need to be piggish and paint modern physicists as absolute louts.
My apologies to modern day physicists. I respect you guys. Your work with metallic hydrogen looks very promising and you are making great strides with that cloaking device but I want to be a little gonzo right now.
In general stories and games take a conservative approach to FTL. If technological advances or levels are part of the IP higher tech levels make you go faster. That's it. There are other areas that could be improved.
Let look at the Traveller jump drives. Tech levels will let you improve your jump range from one to six parsecs (which seems to be the limit but hell they said that about the J-3 drives a few centuries ago.) What if higher tech levels let you reduce the exclusion radius, i.e. you could jump closer from or break out closer to worlds?
TL 9-10 100 radii
TL 11 80 radii
TL 12 40 radii
TL 13 20 radii
TL 14-15 10 radii
Militarily, this will raise hell with your neighbors. If you can jump closer to their installations you can attack them with less warning time. As the jump radii drop a system's defenders move closer to their mainworld and other juicy targets. Jumping closer to a gas giant is a huge advantage because your fleet will reduce transit time for refueling and their vulnerability while playing 'Who's High Guard'?
Economically it provides an advantage to traders. Less jump radius means a faster insystem flight which means more time to spend hunting up cargoes and more trips per years (maybe per month at higher levels). There's also less opportunity to be jumped by pirates.
Of course for player characters this means that the window between THEM learning what you did and you doing a quick escape is much smaller. This usually means less holes in your ship after the climax.
On the other hand ships with less advanced drives than their world of departure or destination will spend time away from defense satellites and patrols and are prey for pirates jumping in from further out. Do these worlds spread their defenses out to accommodate merchants or keep close and defend their worlds better?
What about jump anchors? This is non-canon but some referees play with a minimum mass necessary to drop your ship back into normal space, usually Mars-Earth sized (though I may be remembering Niven's The Borderland of Sol).
TL 9-10 Stellar Mass
TL 11 Jupiter Mass
TL 12 Earth Mass
TL 13 Mars Mass
TL 14 Lunar Mass
TL 15 Asteroid Mass
Using this rule creates choke points in your star systems based on the mass of worlds. A TL 11 world would only be able to attack its enemy by jumping into the system near a gas giant or around the primary giving them some warning. Their TL 12 enemy could appear directly over their world however and attack much more quickly.
A secret asteroid base is also less likely to stay secret when both sides achieve TL 15 and can jump right to it rather than make a long trip from the nearest planetary mass.
Jump fuel seems to be used according to the distance jumped without regard for tech level. More efficient drives might let you make more jumps than lower tech ships bypassing the defenses of a neighbor, needing less tankers and fewer fuel stops.
this assumes there is only one way to access FTL space. And that there are either no FTL communications or everyone has them. A culture that develops FTL radio before its neighbors has an immense advantage.
Who's to say an Empire will be the first to make these technology leaps? When they are evolving and progressing a Polity will be able to use superior drives to great effect. It will project its forces further and quicker than neighbors and grow swiftly as they fall into line or are conquered.
Polities age. Other newer polities (not big enough for a capital 'P') are out there and working on jump drives as well. They might benefit by an unorthodox approach or a better system to motivate and fund research ("Develop the new drive or your kitty suffers!") Maybe the Polity has just stopped trying or the researchers found better jobs in the video game industry.
A 'barbarian' state that gains one or more advantages is likely to use it fast! Consider, as soon as the Polity learns of it they WILL drop everything and begin their own research to close that gap. Or they will panic and orchestrate an immediate attack since their only advantage lays in overwhelming the barbarians. Or they could negotiate with the barbarians who would either give them their secrets and trade their shot for glory, or refuse and be back to square one.
Every day the barbarians wait the Polity will be shoring up their defenses or readying their attack. Regardless the situation is unstable and given the Polity is much bigger their only choice is to attack with everything they have and hope to kill the Empire while it is still in disarray, and behind technologically.
So the Empire will fall, their Fleet swept aside. Many SF Empires end that way. They never tell you the 'barbarians' are more advanced than the 'civilized men'.