  We propose a framework to prove Malle's conjecture for the compositum of two
number fields based on proven results of Malle's conjecture and good uniformity
estimates. Using this method we can prove Malle's conjecture for $S_n\times A$
over any number field $k$ for $n=3$ with $A$ an abelian group of order
relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively
prime to 6 and for $n=5$ with $A$ an abelian group of order relatively prime to
30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$
in its $S_9$ representation, whereas its $S_6$ representation is the first
counter example of Malle's conjecture given by Kl\"uners.
