Yes, Ford-Fulkerson always finds the cut that is "closest" to the source. See this question for a formalization of what is meant by "closest".
A graph can contain exponentially many min-cuts, so beware that any procedure to enumerate all min-cuts must take exponential time in total in the worst case.
Based on what I've read, there are output-sensitive algorithms to enumerate all min $(s,t)$-cuts. After a single maximum flow computation, the running time is $O(n)$ per min-cut that is output. Details can be found in the following paper:
Jean-Claude Picard, Maurice Queyranne. On the structure of all minimum cuts in a network and applications. Combinatorial Optimization II: Mathematical Programming Studies, volume 13, 1980, pp.8--16.
Unfortunately, there does not seem to be a non-paywalled online version of the paper.
(See also here for a related question.)
In contrast, if you want to find all min-cuts of an undirected graph, rather than all min $(s,t)$-cuts, you can use Karger's algorithm to do that efficiently (in polynomial time).