I consulted two editions of Hermann Hesse's poems:
- Die Gedichte, edited by Volker Michels (Berlin: Suhrkamp, 1953; reprinted in 1977 and again with an afterword and dates in 1992).
- Die Gedichte, Volume 10 in: Hermann Hesse: Sämtliche Werke (Berlin: Suhrkamp, 2002; this is part of a 20-volume edition of Hesse's works, edited by Volker Michels).
In both editions, the poem "Frühling" is part of a section titled "Aus den Jahren 1899–1902" ("From the years 1899–1902"); in the collected works, Volker Michels dates the poem to April 1899. It is not entirely clear from the information in these editions whether the poem was part of the volume Gedichte that Hesse published in 1902. Hesse wrote around 1400 poems, 800 of which were printed in 15 collections he published during his lifetime.
By the way, Hesse wrote several other poems titled "Frühling", namely in 1907 (first line: "Der laue März und der feuchte April"), another one in 1907 (first line: "Es fahren leise junge Wolken durchs Blaue"), yet another one in May 1907 (first line: "Wieder schreitet er den braunen Pfad"), one in 1909 (first line: "Seit wir, wilde Knabenhorden") and one on 25 March 1915 (first line: "Am Waldrand tränen die Knospen"). So someone who doesn't check the first lines may get the dates mixed up.
The first collected edition of Hesse's poems, Gesammelte Dichtungen (in 6 volumes), was published in 1952; since this is several years after the Four Last Songs, this edition can't have been Strauss's source. The only other volume of Hesse's poems from the years 1945–1962 was Stufen, which contains a selection that Hesse made at the request of his publisher. This selection was meant to replace the much bigger volume Die Gedichte from 1924, which had gone out of print. This 1924 edition already contained a section titled "Aus den Jahren 1899–1902" (see above).
Hesse also wrote three poems titled "September"; the one that Strauss set to music dates from 23 September 1927. "Beim Schlafengehen", the third of Strauss's Four Last Songs, dates from July 1911.
So James McColley Eilers is definitely mistaken when he claims (emphasis added):
between 1945 and his death in 1962 he wrote some 50 poems, among them the three that Strauss set to music.