So far there isn't really any way to increase leadership in the early game, right now the main method for leveling leadership is by creating an army, which you cannot do unless you are a vassal, or have made an independent kingdom,though there's some really useful skills in the leadership tree(like the disciplinarian perk) there's a few methods I can think of to increase leadership that would probably work;
recruitment - I swear this already increases leadership in the vanilla game, as I am fairly certain I leveled up leadership like once through recruiting new soldiers, but it's super rare, so increasing the exp gain would probably work.
upgrading troops-not in vanilla(as far as I am aware), but leveling a tier 1 to tier 2 to tier 3 etc could give leadership exp, this would tie in pretty well with the leadership perks too.
renown gain-also not in vanilla, wanted to keep it separate from tactics, so rather than simply gaining exp from winning battles, the amount of exp would be based on renown gained in the battle, so winning against looters would naturally give less renown than winning against an army, so there would be less leadership exp in that situation. The nice thing is this would likely work with quests that raise renown.
morale- the description of the leadership skill says high morale increases the skill, but I'm not sure how high the bar is. I feel like I've managed to get my morale to like 80-85 and there is no level ups for the skill, maybe lower the bar or increase exp gain from this or something?
One thing I'm not sure of is how much exp each of these ideas should give, so some fine tuning may be required. I wanted to try to keep leadership as a separate thing from tactics with these ideas.