You need to push a reference to the sublist onto the bigger list, taking care, as noted, that Perl does not “flatten” them. (Data::Dumper is your best friend.)
The reason why references are needed is that, in the end, all of these data structures are one-dimensional. If @biglist is supposed to contain, say, 5 sublists, then it will then be a list containing 5 entries, each one of which is a reference to its own separate list.
There are, as noted, several ways to build the final structure, and Data::Dumper is quite smart about printing out what it actually sees. Create a tiny stand-alone test case to prove your logic.