What to do when you have small yet additional results to a submitted paper? Suppose I finished work on a manuscript and submitted it to a journal. After X weeks, I suddenly realized I can prove one of the conjectures I left open in the concluding section. It seems that X is small enough (e.g. 1) so that nothing visible has happened to my submission yet.

The proof is not long, and wouldn't be important enough to merit its own paper. Will it reflect badly on me if I contact the managing editor to have the current submitted version updated? The smaller X is, the less annoyed people would probably be. I could just submit to arxiv, but I would like to make the manuscript stronger, and don't have results scattered around giving a hasty impression either. Or is there another solution I'm overlooking?
As Marc says in his comment, the best thing to do is probably to incorporate the new proof into your revision, after you get reviewer feedback. A small extension not worth its own paper is unlikely to make a difference between reject and revise. If you send it now, it may annoy the editor and reviewers. Also, the reviewers will likely be just fine with more new material you add in revision.