How do I raise the Terran reputation after the main plot?
Factories in heretics end will increase terran rep but it is unfortunately a very slow process.
Killing pirates / xenon in segaris will give more rep but it is very unlikely you will find enemy incursions in this area.
Ramming argon vessels in their home sectors will also slightly increase terran rep but at the expense of lots of argon rep.
Do missions in the Aldrin smaller systems next to the massive main Adrin system (don't bother doing mission in that system though "too big"), do them in Adlin 2 and Megir, and it will raise your terran rep. You get treated slightly different with rep towards Aldrin bases and ships in their own sectors - than you do in the main Terran sectors. But doing missions in the Aldrin systems raises your rep with Terran also.
Just a note, you won't be able to kill Xenon to raise your rep in Segaris playing X3AP. They don't invade the system like they did in X3TC. They don't even enter it.
Finally I got some recognition with the major Terran factions.
Having -3 (Terran Insurgent & ATF Known AGI Sympathiser, both 60%) I started attacking small ships (M5 and below) appearing in "Heretic's End" piloting a Mamba. You can probably have an easier time using a M2 destroyer or several M6 escorts but be aware of Terran response units - run away and don't engage and jump away if they pop in, to save Terran reputation and your ships obviously.
This petty tactic does seem to increase Terran and ATF rep by about 1% for each fighter ship kill (note that Argon ships will pop in every now and then in the above mentioned sector). After doing this for a while I got to about 78% of both Terran ranks (Insurgent & Sympathiser) and got an "Escort Convoy" average mission from a heavy Terran heavy transporter in "Heretic's End" by chance. I was able to pass the mission with lots of guns (a Boreas, two Centaurs and a Mamba) and about a dozen save/loads (keep to your M2 close to the escort and don't use Ion Disruptors). The mission's route was from the middle of "Heretic's End" to a mining asteroid base in "Asteroid Belt", so be well prepared if you get an escort one. Upon completion I ended up with "Terran Marshall of Mars" at 55% and "ATF Venus Access" at 55% reputation for both Terran factions, which seems pretty good.
Note that I started with the highest "Argon Hero of the Federation" reputation status and ended with "Argon Trusted Ally" at 28%. This took about 5+ hours (if not even more), so don't expect a quick solution (unless you can optimize this). So have a lot of Argon reputation before attempting this solution and make sure to set the Argon faction as enemy only to the ships involved in the mission. However, once you achieve this, the X3 world is yours, Argon rep is easy to gain if you started with a full bar.
Using X3: Albion Prelude v3.1