Each pack of hot dogs contains 10 hot dogs (no buns), but each pack of hot dog buns contains 8 buns.  Phil buys a number of these packs for a barbecue.  After the barbecue, Phil finds that he has 4 hot dogs left over.  What is the SECOND smallest number of packs of hot dogs he could have bought?
Let $n$ be the number of packs of hot dogs that Phil bought.  Then $10n \equiv 4 \pmod{8}$, which reduces to $2n \equiv 4 \pmod{8}$.  This congruence tells us that $2n = 8k + 4$ for some integer $k$, or $n = 4k + 2$.  The second smallest positive integer of this form is $\boxed{6}$.