In a nutshell, the strategy is to test cousins.  That is, you test other male patrilineal line descendants of the subject's alleged earliest ancestor.

If you have a suspicion where the NPE or mutation occurred, you can start with a cousin connecting at that generation, but if you have no clue, it really doesn't matter in what order you test them because it's going to be purely a matter of chance whether you pick the right one, or not.  The difficulty, of course, is in finding those cousins and getting them to agree to be tested.  If you want a speedy resolution to the matter, you will most likely have to pay for the testing.

The easiest test subject to locate may be a brother, the father, an uncle, or the paternal grandfather, if living.  Otherwise, the place to start would be with a first cousin.  If you match your first cousin (or a paternal uncle), you know the NPE or mutation did not occur in you, your father, or your paternal grandfather.  For most individuals, that knowledge relieves the social pressure to keep an NPE secret, and I recommend keeping it secret until you have this knowledge.  Some people take the news of an NPE in stride, while others find it devastating.  The further back the NPE, the less likely it is to upset anyone.

On the other hand, the further back the NPE, the greater the number of descendants affected, so the more important it becomes that these NPEs be resolved.  In other words, the further back the NPE, the more important it is not to keep it a secret.  Someone else will discover it eventually, anyway, so the sooner it's resolved, the better.

And the bottom line here is:  every male needs to be Y-DNA tested because no male can be certain of his patrilineal line ancestry until he does.

As the issue has arisen because the subject doesn't match some other alleged paper descendant of his progenitor — or is matching a family with a different surname — filling in this table can begin with placing the cousins already tested into the table, then proceed by filling in the test results of other cousins until you find the generation where the answer changes from Yes, they match the Subject, to No, they match someone else.