Some Neat Reputation Tricks
One of the best ways to get an improvement in a faction's attitude is to fly missions for one of its friends. This isn't going to turn a faction from your worst enemies into your best friend (unless you do it a LOT), but it's great for getting a faction up from hostile to neutral to get them off your back, or for getting a faction just over the line into friendly so you can take missions for them.
Let's say you're hostile against the Corsairs, and want to get them to neutral to get them off your back. In empathy.ini, we see that the ONLY two factions that are significantly friendly with them are Hogosha and Junkers. (The Corsairs hate a LOT of people, so trying to bribe them will cause rep problems with a lot of groups.)
Let's look at the enemies of those factions. The Junkers hate Xenos - perfect! Go to a Junker base (I used Beaumont in Texas), and look at the mission board. Missions against the Xenos are available, although you might have to reload a few times to find one among all the missions against the Bounty Hunters.
Each mission you fly will improve your rep with the Corsairs by about one-fifth of a box. You can knock off missions in 4-5 minutes each, flying a powerful ship against level-2 Xeno ships. After about an hour, you can have the Corsairs up out of hostility and neutral to you. (Add another hour or so if you want to get them up above zero, and then you can go fly missions for the Corsairs themselves if you want to get them to friendly status to, say, buy a ship.)
What side effects does this have? Well, according to empathy.ini, the only faction that the Junkers significantly hate is Hogosha. But - get this - the Hogosha also hate the Xenos. Running missions for the Junkers will make Hogosha somewhat angry, but you'll make up for that with all the Xeno kills!
There's also the neat side effect that the Junkers also have good empathy with another faction that's hard to get in good with: the Red Hessians.
All in all, flying missions is the finely precise scalpel to adjust your reputation with different factions exactly as you need. After an hour of doing that for the Corsairs, and half an hour flying missions for friends of the other two groups that hated me (flew for Unioners to please the Gaians, and Gateway Shipping to please IMG), I've got my rep neutral or better with everyone except the Xenos, and I can go explore or run cargo anywhere I want with nobody bothering me.