The residual capacity of an augmenting path is the minimum capacity of one of its edges. Thus, we are interested in finding a maximum capacity path from $s$ to $t$. We can perform this computation by using a maximum spanning tree algorithm, which is just like a minimum spanning tree algorithm with all the weights multiplied by $-1$. The path from $s$ to $t$ in this tree will be a maximum capacity path (using a similar argument used to prove the important fact about minimum spanning trees).