Its always good to read docs anonymous monk above mentioned important docs. But to push to 3-dimension, it should go like this

# push @tmp into 3rd dimension
push (@{$AoA[$i]},@tmp);

# push @tmp cell to 3rd dimension
# testing for proper info, not tested
foreach (@{$AoA}) {
  next if (defined($_) && ref($_)!="ARRAY");
  push (@{$_},shift @tmp);
}
[download]

Remember to make sure the cell of array you are pushing into is either undefined or array reference.