To be clear, even if you're moving at 1000x the speed of light, and have figured out a way around relativity so that you don't have time dilation (and so you can, y'know, go faster than the speed of light), galactic distances are still so vast that interstellar traffic is largely not feasible. Our galaxy--one of billions, trillions, or more--is about 2M light years across. Going all the way across the galaxy at 1000x the speed of light would still take 2000 years.
Milky Way is much smaller than that. The Andromeda galaxy is 2.5M light years away. Maybe that's where you got 2M from. The Milky Way is about 100,000 light year diameter.
If we somehow figured out how to travel at 1000x speed of light, I'm sure we'll be able to increase that number to 1 million, 1 billion, and beyond. Until we just stopped using "speed of light" as a frame reference because it would be a completely different method of traveling (like instantaneous through a wormhole or something).