I have the vacation time to take a weeklong getaway a month after next so do you want to go somewhere with me?
The next month (来月) isn't enough... one more next month, please. Let's do it again. That month is the month after next.
This is a jukugo word, which usually means on'yomi readings from the kanji. If you know the readings of your kanji you'll know how to read this as well.