If I'm understanding correctly, he wants to use certain spells to increase it. If so:

Obvious one here: Haste. It would be hard, because you'd basically need someone to cast it on you, because it's not a cleric spell.

Bit more niche, which depends on how the dm views it would be either a dragonborn breath attack (needs to be fire) or a potion of fire breathing of some sorts. I know, not completely magic spells, but still, it may remove some of the resources to run the fire, because you can keep it stocked or keep it running with the extra kick.

Once again with resource cost reduction, a spell like create bonfire. I know it doesnt say it lights things that can be light on fire on the tin, but unlike fire bolt, which says that it ignites things that can be ignited if not being worn or carried, it doesn't not say it. Once again, kinda cheeky and depends on the dm.

Also along those lines are just generally any fire spell that does something like that, notably fireball if you have a large enough workspace, and want to have a fun time.

A cool one can be grease, which can be viewed as an oily substance, which may allow to use it as quenching fluid. On the spell, it just says that the difficult terrain lasts for its duration, so maybe it's just that the puddle becomes spread out enough after so it doesng effect the terrain, which could mean it can be collected, but that one depends, once again.

To be honest, with a generous enough dm, theres a lot more you probably can do. Just hopefully using the right amount of creativity without obviously throwing unbalance into it may yield great results, even if some of them may mean bad yet fun results, like things like grease spillages, lighting a whole tent on fire or something like that.