Personally, I would think that without the Missouri Compromise Line, you'd see butterflies that may or may not affect there being that land strip in the first place.
Considering the line heavily disfavored the South (compare how much land ended on each side of the line), without it, they might not be as eager to expand west to secure more land for states if they can just work on the land they already have, with no arbitrary line that say they can't. While there would likely still be sentiments of expansion, you'd have removed one of the reasons of the South's support of it, and might now lean to a mindset like the Whigs. That is, to focus on what they already had.
Anyway, assuming things go as OTL, that strip of land would might as well remain part of Texas. I don't really see much note to it, if the line isn't there to tell Texas it can't have that small strip of land that happened to end up north of it in OTL. Heck, they could perhaps even keep it further north to the Arkansas River. But well, now that, I don't know.