This is really a comment, but it's getting a bit long for the comment box.
I want to point out that in addition to what you call the "greedy process," there's another obvious attempt, which could be called the "stingy process."
Let a target $\mu$ be given. There's basically one rule, namely don't squander unused space unless absolutely forced to. More formally, if I have already chosen points $0\le x_1, \ldots , x_N\le a$, then these define $N$ intervals $I_j$. I must now place the next point such that the minimum distance is $\ge\mu/(N+1)$ (I will keep the whole sequence $\ge\mu$, not just the $\liminf$). If I have an interval of length at least twice this distance, then I put my point inside such an interval (let's say inside the smallest one that works and I will also make one distance equal to $\mu/(N+1)$, though that might not be optimal). If not, then I grudgingly add an interval of that length to the right end of the current configuration.
This gives me intervals that get subdivided, and every once in a while a new interval gets added at the current right endpoint. The whole procedure will be a success if the limit of these right endpoints is $\le 1$.
I fooled around some with this; let's take $\mu=1$. Then the first few points are $$ 0, 1/2, 1/2+1/3, 1/4, 1/4+1/3+1/5, 1/2+1/6, \ldots $$ It seems that $\lim a_N =1/2+1/3+1/5+1/7+1/11+1/14+\ldots \simeq 1.4$. If this is correct (but I'm not making any strong claims it is), then it means that we can get $\mu\simeq 1/1.4\simeq 0.7$, which is better than the golden mean.
In any event, it should be easy (for someone more computer savvy) to do the numerics and get a more reliable estimate.