Unfortunately, BattleTech has ignored the 3D aspect of interstellar travel until it's too late to fix it. (The chance to fix it passed in about 1986.) Entire tracts of history in the House Sourcebooks and Handbooks, and the Clan sourcebooks, have been written as if centuries of conflict were crawling across 2D interstellar maps. Clan and House territories are profoundly, utterly intertwined with 2D maps. There's no way to wave a wand and say, "the Inner Sphere is X light-years thick" or "there's Y light-years between the highest and lowest stars in the Inner Sphere," because then all those 2D conflicts need rewriting.
That means the canon, rules- and fluff-backed answer to your question is, "...hey, look, it's Elvis behind you!"
[runs away before the question can be asked again] ;)
If you'd like to involve the third dimension into your home game's interstellar travels, I'd give the Inner Sphere the same Z dimensions as its X and Y dimensions. It's called a sphere, so it should be as deep as it is wide and long: about 500 light-years radius. But that's a personal opinion, not anything backed with canon.