'Warm' or 'cool' papers are described thus precisely because they affect the look of the print.
Because the result is purely an aesthetic one what you need to do is experiment and learn which sorts of papers give the result you like.
A particular paper may just never give you what you want, yet another paper may hit the mark.
What is acceptable, and desirable, is absolutely subjective.
Most expert printers appear to use only a very few surfaces, a gloss or semi-gloss, matte, and maybe a canvas type as well - so only two or three papers. They do change occassionally, but a lot of testing and comparison is involved.
It appears that the 'cool' paper gives you what you like but the 'warm' paper doesn't - it is possible that you may never warm to the 'warm' paper and that is absolutely fine.