Yes, the boost exponent is the hyperdrive. But you are just pushing buttons without understanding how the SE hyperdrive works.
When the hyperdrive autopilot first starts up, it aligns the velocity vector with your target. It may boost you in a direction not to your liking, but you are talking about a few hundred kilometres at best, compared to 6 million light years? How far out of your way is that?
Plus you are picking very distant targets which would normally takes the typical SE ship days, weeks, or even months to get there. You can crank up the real space velocity to get there quicker, sure — even use black holes as a slingshot — but then you also have to slow down at the other end. If you are under thrust all the way to your target, half of that should be decelerating.
Pick a closer target. Watch the SE autopilot do its thing. The velocity vector is the direction you are travelling in space. You have to do engine burns to bend your path towards your target. I don't know how simpler I can put that.
I actually have no idea if I made it to my destination (I was within 3 hours), because I fell asleep and the computer automatically shuts off in the middle of the night. It probably made it there but I noticed there were no warp effects going there, which means I probably wasn't using the warp properly. So I read more from the manual and decided to try it your way. First let me completely outline my method so it's easier to figure out what I might be doing wrong.
1. Start Space Engine
2. Use Find Object from Left Control Panel to Find Earth and Go There
3. Use Find Object to Find NGC 300 and Select Ok
4. Use Lower Left Control Panel to Center NGC 300 on my screen.
5. Use Spacecraft Manager from Left Control Panel to Build New Spaceship.
6. Take Control of New Spaceship.
7. Switch New Spaceship to Warp Mode.
8. Go back to Find Object in Left Control Panel to Select NGC 300 again.
9. Use Lower Left Control Panel to Center NGC 300 on my screen again.
10. Set Main Engines to 100 to start movement.
11. Set Boost Exponent as high as it will manually go (11.637).
At this point I had my first error (field unstable, turn to target, etc., even though the target was centered on my screen.)
12. Rotate to Target.
13. Hyperjump.
I definitely have the warp effects back and I love the look, however, I'm not sure I'm headed in the right direction even though I had NGC 300 completely centered when I built the new space ship. Not sure where I went wrong or what else I could've done because this happened 4 times in a row now. But I'm hoping that you're right and it's just going to take the scenic route to get there.
Here's a screenshot with some nice warp visuals.