"Minimum" maximum flow with extra capacities
The smarter Naive approach of extending a solution to the Maximum Flow problem to a similar graph with at least as much capacity is to only add flow to augmenting paths that you'd find from the Residual of the initial graph. And if no augmenting paths exist, then terminate. This is known as the Ford-Fulkerson Algorithm, and if you were to have run it on the a graph of $E$ edges initially, this would incur a runtime of $O(E\cdot f^*)$, where $f^*$ is the maximum flow in the graph.

In case you haven't heard of some of these terms, or the Algorithm, visit: http://en.wikipedia.org/wiki/Ford-Fulkerson_algorithm