I believe the following achieves

8 weighings

First,

split the 30 coins into 6 groups of 5, and weigh each group. Only the lightest two coins from each group are candidates for second lightest overall.

Then,

weigh the 5 lightest coins from 5 groups, and sort those groups in that order. This eliminates the 3 heavier groups.

Finally,

weigh the remaining candidates: the two lightest coins from the first group, the lightest coin from the second group, and the lightest two from the unweighed group. These include the two lightest coins, so take the second lightest among them.

I believe the following achieves

8 weighings

First,

split the 30 coins into 6 groups of 5, and weigh each group. Only the lightest two coins from each group are candidates for second lightest overall.

Then,

weigh the 5 lightest coins from 5 groups, and sort those groups in that order. This eliminates the 3 heavier groups.

Finally,

we know the relationships below, with arrows pointing from a lighter coin to a heavier one. Weigh the 5 coins marked in blue: the lightest two from the first group, the lightest one from the second group, and the lightest two from the unweighed group. These include the two lightest coins, so take the second lightest among them.