I have discovered that, if you have good enough standings with a faction to buy/sell, but your Alliance has Abhorrent or otherwise negative standings, you can fix the Alliance standings by doing the following procedure:
1) Arrive in system of a faction in a personal ship, or drone. The system must have something to buy or sell.
2) Pull up right next to the station.
3) Found a new ship, if you are in a drone. If you are in a personal ship, skip this step.
4) Open the trading window of the station (let it be trade goods, Resource Traders, Equipment Docks, etc).
5) Use your mouse to click the Player button.
6) Click on the Alliance ships list, and click "Transfer Ship".
7) Make your purchases, before closing the window. Note that the prices will spike/drop (buy/sell) once you make your first purchase, due to your initial standings.