Effect of increasing the capacity of an edge in a flow network with known max flow
I am assuming that you are given the flow on each edge which corresponds to the maximum flow for the graph $G$. So $f_e$ is the flow on edge $e$. 

I am also assuming that all the capacities and flows values are integral.

Now given this information, capacity of an edge $e$ is increased by 1. Therefore, the mincut value can increase by at most 1 implying that the maxflow can increase by at most 1.

Thus, find the residual graph and check if there is an augmenting path $P$ from $s$ to $t$. If such a $P$ exists, then increase your maxflow by augmenting this path, otherwise the current flow is still the maximum.