Spartan kings did try to reform the state into having a larger citizenship manpower pool to draw from. That was done mainly through the redistribution of land under the kings Agis IV, Cleomenes III and King Nabis to an extent.

King Agis IV attempted to redistribute the land in equal "kleroi" (land plots) as well as supposedly wanting to abolish debts and give citizenship to foreigners. According to him, problems faced by Sparta could be resolved through equality and restoring the citizen body (anaplerosis), which had been consistently depleted.
He also wanted to include his own land along with his friends' and family's land in the distribution (anadesmos ges). It is also interesting to note that the majority of land was owned by noble women. Agis' mother, supposedly objected to the redistribution of land but Agis convinced her to participate. Agis' reforms led to his death however, after a coup in 242B.C. His enemies waited for the abolition of debts to go through and then moved against him. Agis along with his mother were executed.

Equal distribution of land along with banning loans was within the Spartan "Lycurgan" constitution (that had existed for centuries), so these "reformer" kings did not do something out of the ordinary or something that is not within the Spartan constitution (except maybe adding foreigners as citizens).

Cleomenes III was a very important Spartan king. To Cleomenes, the existence of both poverty and wealth was a problem plaguing Sparta. Cleomenes claimed Sparta to be "a city of equality", not only economically but socially, too.

He married Agis' wife (after Agis was murdered) and continued with Agis' plans. He was successful in murdering the Spartan ephors, and the redistribution of land finally went ahead. The historian Polybius is hostile to Cleomenes and his reforms (I'll explain why a bit later on) but, overall, Cleomenes reforms were a success. Cleomenes' mercenaries were also adding to the Spartan distribution of land. One interesting thing to note is that Cleomenes also allowed land-plots for his enemies too (not only his friends/ allies), after exiling them.

Sparta was able to field a large-than-before army, an army that was supposedly well-trained and for the first time Spartans were trained to fight in the Macedonian phalanx tactic.

With money taken from the exiled rich, Cleomenes' Sparta began to go on the offensive again, capturing the important city of Corinth, and defeating their biggest rivals (the Achaean League) multiple times. Cities in the Peloponnese willingly sought out Cleomenes due to his abolition of debts and land-redistribution scheme. Cleomenes was an able general, outsmarting the famous Aratos of Sikyon.

Cleomenes' downfall were (indirectly) his reforms, in a way. The Achaean League (and specifically Aratos) sought out the help of the Antigonid Macedonians, requesting help against the Spartans, the invasion of a large Macedonian army led to Cleomenes' defeat and flight to Egypt after the Battle of Sellasia. I say that his reforms led to his downfall because it is said that Aratos sought out a Macedonian alliance "in order to avoid the Spartan barley bread and short cloat" along with avoiding "the most most dreadful of evils, namely abolition of wealth and putting poverty aright" (Cleomenes 16.7).

Polybius is hostile to Cleomenes, calling him a tyrant but doesn't mention Cleomenes' reforms, as that would be contrary to him being a tyrant. Polybius is likely hostile to him because Cleomenes forced the Achaeans (Polybius was an Achaean) to call the Macedonians for help as well as defeating Aratos (Polybius' role model).

If you want to read a beautiful poem regarding Cleomenes' flight to Alexandria, read "Come, O King of the Lacedaimonians" by C.P Cavafy.

References:

Polybius and the Reforms of Cleomenes III, B. Shimron
Agis, Cleomenes and Equality, Alexander Fuks
Cleomenes III and the Helots, Africa, Thomas
Plutarch, Parallel Lives