Well Blackwood, there is no magic formula to gain loyalty as quickly as you want.
The fastest way is to sac heat to your protector.
There is an option in MD Shop to increase briskness for 1 hour. You can buy that and walk, then sac heat.
You can also buy crits and then sac them. You will gain heat for this and again..sac it.
Also fight [spoiler](lose the fight and you get more heat but you lose more ap)[/spoiler]. Gain heat, sac it.
If you attack ally members you can gain up to 15 loyalty points in one fight but you have to meed some conditions to reach that amount. It's up to you how you decide to do it.