If the order doesn't matter, you could sucessively chop off stuff from the end.  This prints the same as your code, but reversed:

my $_=$a;  say while s|[^/]+/*$||;

__END__
/aaaa/bbbb/cccc/dddd/
/aaaa/bbbb/cccc/
/aaaa/bbbb/
/aaaa/
/
[download]

But if order does matter (e.g. for mkdir purposes), you can of course also say

my @dirs;
my $_=$a;  push @dirs, $_ while s|[^/]+/*$||;
say for reverse @dirs;

__END__
/
/aaaa/
/aaaa/bbbb/
/aaaa/bbbb/cccc/
/aaaa/bbbb/cccc/dddd/
[download]