I'm a PhD student in Mathematics. Last year I submitted a short paper in enumerative combinatorics to one of the top Combinatorics journals, which sought two referee reports on my paper.
One of these reports was exhaustive, went into the complete details of the then somewhat poorly written paper, suggested a lot of changes and asked for illustrative examples that were missing then, and said very positive things such as "the paper deserves to be widely read", "the method is very ingenuous and should be influential", "the revisions though tedious are well worth the effort for the many readers the paper deserves".
However the second referee thought the paper was only "somewhat interesting" and said there was a lack of examples and explicit constructions, and the totally new theorems lacked sufficient appeal on their own to get published in this journal, although the purpose was to give new and hopefully interesting proofs to old results as well. It didn't seem like the second referee read beyond the introduction, which would be fine with me if this was the only feedback, and I would have immediately tried a new journal.
The editor requested me to make all the exhaustive changes that the first referee asked for. I took a while to do this, with several explicit examples added that also partly addressed the second referee's concerns, and resubmitted. However now the second referee had more general negative comments and opinions about the purpose of the paper, whereas the first referee suggested acceptance. The editor rejected the paper.
Now I again rewrote the article to address some of the further concerns of the second referee, and in my humble opinion the article remains just as interesting as I thought it was in the beginning.
Would it be pertinent to contact this same journal again, with these few further changes, or would it be proper to submit to a new journal of similar stature and request the editor to contact the old referee who substantially read the paper and eventually recommended acceptance?
Looking at a few other questions, it seems it is standard practise for referees in Mathematics journals to consent to having the referee reports transferred, but would it be considered improper for an author to suggest contacting the old referee, not knowing if the old referee consented in a form to having his report transferred?