Been in a full-time, exempt, temporary position

That position is coming to a close, but have been asked to step in to a different, also temporary, part-time position - details (employee vs contract, hours, $$) have not yet been negotiated, yet you've agreed to the idea in theory

You are well-liked and respected, but you don't necessarily have the specialization of the person leaving the part-time role - it's more a function of the timing - that you happen to also be exiting and are therefore available, are well liked, have the flexibility to continue, and can totally do the work well, specialization notwithstanding. the ideas is that you will steward the work in the interim while they work to hire someone with more specialization permanently. A couple months

You know what your current salary boils down to in terms of an hourly. You know what the previous part-timer got paid as their hourly on a contract basis. Unsurprising, their hourly rate eclipses yours should it continue unchanged

You're considering suggesting terminating employment and continuing as a contractor, and an hourly rate that falls exactly between what you were otherwise making, and what the previous part-timer was making. This represents a meaningful increase to your hourly, but also a meaningful decrease in the hourly they would otherwise be paying this other person.

Relations are smooth and you want to keep them that way. You're also about your $$. Anything special in the way you propose this idea? Do you think it's a reasonable idea to propose? Unfortunate that these details weren't ironed out at an earlier date, but everyone is touchy about discussing money plainly and has been avoidant. You're halfway committed regardless, and they totally need you regardless, so the negotiating "position" is about even in that way