@CountPeter 's proposed reforms relating the expanding the Spartan manpower pool and getting over the cultural hurdles in the way of them was what I was addressing, which will I admit inevitably require an increase in the amount of territory/productivity directly under Spartan control in order to sustain (though this would be the case for literally any state using the citizen-soldier landowner model). Though, I'd argue primogeniture dosen't fundimentally solve the problem as you STILL end up with a large number of younger sons who won't be able to maintain their citizen status in that case unless they can obtain land by some other means (Which means you, again, have to get more land under direct Spartan control), and that in order to expand Spartan power you need... well, a larger Spartiate population. Your proposal to expand its strength via expanding the League without that power directly increasing the city-state won't work for establishing Spartan hegemony as such an expanded League would inevitably see Spartan influence dilluted to the point a collection of the other members will be able to usurp dominance from her, and if the Spartiate are divided in small groups among much larger groups of allies they're going to be spread way too thin for the city leaders to enforce their will if their allies decide they aren't fond of the policy being dictated. Athens could pull it off with control of the purse strings and the fact she had a foot on the seaborn windpipe of prosperity.
If anything, polygamy would be a great way to justify a change in inheritance law as well, as the odds of having no male heirs would shrink and the maintenance of the estate out of the direct control of a single wife (to maintain the others) would help prevent the rise of the Spartan Heiresses and with it help styme the excess concentration of wealth