Many of the largest sized suitcases will fit your cue case in diagonally. Almost all of the largest suitcases are near the airline maximum OAL of 62" (Length + Width + Height = 62"). Even though the large size suitcases are all similar volume and OAL, they have slightly varying shapes. The ones that are the longest (or tallest depending on how you want to look at it) are going to be most likely to fit a case the best. If one is 28" long and another is 30" long, the 30+" long one will be more likely to fit a case diagonally.
Another hint is that if your cue case is just short of fitting diagonally, a soft sided suitcase (instead of the hard sided ones) will often deform just enough (into a very slightly diamond shape instead of perfectly rectangle) to let your cue case fit in anyway.
They also make super long suitcases as well as wheeled duffels that work well. I would suggest either a large suit case that will fit your cue case diagonally, or something like the very long rolling duffel below which is convenient because it has wheels so you won't have to lug it everywhere on your shoulder, plus your cue case will fit in it easily without even having to be diagonal, and there should still be plenty of room left for all your clothes and everything else you are taking on your trip. Shop around and you might be able to find a good deal on something like it somewhere.
https://www.rei.com/product/878305/rei-co-op-wheely-beast-wheeled-duffel-34
This same type thread pops up every year around this time and the most popular answers always seem to be a large duffel bag or a bat bag as I recall.