But how about end of January / beginning of February? Will there be enough snow?
That is the sweet spot for maximizing the combination of decent coverage and pleasant snow surfaces at Jackson. Powder, as Tony states above, is not predictable more than a week in advance. For what it's worth January has the highest average snowfall (12% over December, 22% over February) but that doesn't mean much when the standard deviation of monthly snowfall is 46% of the average. Thus one's advance planning should be based more upon what to expect in the ~70% of weeks where you do not see a lot of fresh snow.
President's Week is a rough demarcation line. The week itself by reputation tends to be busy and Jackson's lift system though improving does not handle big crowds as well as many places.
Each week you go past President's Week increases the odds of difficult surfaces due to the predominant SE exposure. This is particularly true for the amazing terrain of the Lower Faces. Midwinter the snow is often well preserved top to bottom by chronic temperature inversions. The inversions are rare after mid-February as the stronger sun breaks them up. You might want to read that March 2015 report and comments in more detail. I was not there but I got several off-line e-mails noting that the variable conditions were quite challenging for skiers below Tony's expert skill level. Liz and I were at Jackson January 22-25 of that same season with excellent conditions.