3 months
suppose the track gets released on 1st January
usually promotion time depends on popularity of group, but lets take a moderately-well off group who is still rising as the average for now
the promotions, only including awards, performance videos, behind-the-scenes, M-countdown performances should be for a month at least for some good hype, which will end by around February 1
Next, some appearances in variety shows and small content, with the idols resting for a month or so ending by March 1, basically giving the song some time to chart higher
last month for them to finish off the minute details regarding the release, filming the music video if needed etc. with the teasers emerging 1.5 weeks before actual release with solo teasers released 2 weeks before as well, ending with the release of the second single on April 1... also, if needed, different versions of the teaser, if it has different themes, can also be released, might add to the hype
the time between the 2nd and 3rd month can be shaved off by 1-2 weeks, but that is the max for a good hype around it
tbh, this only applies to groups rising in popularity, more so than domestic popularity, but it differs mostly, based on the group and its style as well
I knew it :)