To avoid answering several of these, I am going to lay out my favorite theory behind them. Not everything in this answer is fully relevant to this question, but to the others like it here. This is officially redundant. Someone here already said this, but they used heavy notation that seems unnecessarily precise, and off-putting.
Because of the way English usually works, we are tempted to hear the subjunctive statement as an indicative one -- saying that for any time before six, if I started then, I would most likely arrive before noon.
But since this is a subjunctive implication, it means there is an untested or disputed theory behind it, which I currently accept and am basing my deduction upon. That theory is incomplete and might be wrong. If I find some aspect that shoots down my theory, then the statement loses its meaning.
In this case, the idea that I drive equally well at all times seems to be part of the theory. If it fails, the implication does not have any force.
So in this case, if I left at 5:50, I might arrive before noon, because that time before six is one where my theory still holds. But if I left at 5, I would navigate worse. So I have made an even more drastic correction than I would have needed to (strengthening the antecedent), and the correction itself has caused other unstated preconditions to break down.
So this seems to disprove the statement, but in fact it only disproves the unstated theory behind it. In a more complex case, more than one thing might have gone wrong to make the assertion fail. If I have misidentified what antecedent is too weak, strengthening a different one can make it seem like I am being careful, when, in fact, I am just hiding productive lines of inquiry.
When working with subjunctive statements, in order to make real deductions, one needs to know the 'range of all possible worlds under consideration' basically, to know the theory under which the implication would be valid. The problem is that folks usually cannot articulate those theories.
One formal way of dealing with this is to consider 'modal' statements made in the subjunctive as always true. There is bound to be some bizarre theory on the basis of which they would be true. But then we need to require removing (or 'relativizing') the subjunctive before we make deductions from it.
We can truly remove the subjunctive only by turning it into a complex implication "my theory => S", and since we cannot articulate "my theory" by mind-reading the source of the statement, we have to guess as to the completeness of our understanding of the theory.
This approach is called "modal suspension", because proceeding forward from the proposition is "suspended" by the "mood", awaiting further analysis. And it implies that your confidence in the statement depends on whether the theory has been largely identified and satisfied, or whether you believe the source based on relevant experience.