This is an elaboration on MO Question 212550: given $ 0 < 2a << 1 $, how many points can be placed on the unit sphere, subject to the constraint that any two of these points must be at Euclidean distance at least $2a$ ?

Obviously, every point must be allocated at least the area of a disk with radius $a$, so the maximum number $M(a)$ cannot exceed $4/a^2$; however, Fejes Tóth famously proved that the allocated area must be at least that of the circumscribing hexagon, so that: $2 \sqrt 3 a^2 M(a) = 4\pi - S(a)$ with $S(a) > {2\over3}\pi a^2 $.

Then, we have a 1955 (**Edit:** 1951) paper by Habicht & Van der Waerden offering a very contorted construction, and the estimate:
$S(a) = O(a^{2/3}) $. They concluded with an oddly phrased comment, acknowledging their explicit value for the constant is far from minimal, while implying the order of magnitude is tight, i. e. their $O(a^{2/3}) $ should really be $\Theta(a^{2/3}) $.

Then... that's all. Or at least, that's all I could google regarding my question. To wit:

How much is proven, and how much more is plausible, of the behaviour of $S(a)\over a^{2/3}$ for inifinitesimal $ a $? If it is $o(1) $, what is its order of magnitude? If it is $\Theta(1) $, what are its upper & lower limits?

**Edit**: I do not mean Google has no answer to this question; only that they lack relevance. J. Park pointed out in his answer that any explicit construction provides a lower bound; a plethora of such constructions result in $S(a) < C a^{2/3}$ for various $C$'s, some better, some worse than the original one by H.-v.d.W. What's wanting is a public claim of optimality.