Adding new results during review process I submitted a paper to an applied mathematics journal. I have since found a way to generalize one of the results.

Is it acceptable to update the paper during the first review step even if I have not been requested to improve on this particular aspect?

I understand that this would create extra work for the reviewers since they would have to check that part again. However, writing a brand new paper with the additional result would probably cause even more extra work for the review community. I also understand that a dissemination strategy that incorporates some salami slicing might be better for my career, but somehow I am opposed to doing that kind of thing.
Yes, yes it is. Just make it clear when you do so the reviewers can review the new additions. 

That said, if you think the result merit a new paper, make a new paper, and reference the one under review.

However, while I am an early stage researcher my suggestion is: have a high ethos when publishing. It is way better to have 1 very good paper than 3 weak-related ones. Your objective is to show the world new research, do not fall in the paper-mill trap. Your worth is in the quality, not in the quantity.