If on one ticket, a late delivering carrier (in this case SK both delivering & receiving, thus irrelevant) would be responsible for rebooking. But, if the connection is within MCT and the first segment was on time, SK bears no responsibility. Most carriers would accept it and simply rebook, but nothing to force their hand.

It would be helpful if OP would explain where this occurred and the exact timing and circumstances, e.g., why were they late to the gate?

Note that travel insurance will not generally provide any relief if OP was simply late. The reason will matter.