The story is told in Eisenstaedt’s book (2006, Chap. 12) or papers (1982, 1987, 1989) (also jstor).
Briefly he says that, while Schwarzschild and Droste’s original works (1916, 1916, 1917) ignored the region $r\leqslant 2m$ (Schwarzschild actually used $\smash{\tilde r=(r^3-8m^3)^{1/3}}$ > 0 as his radial coordinate), other early papers like Hilbert (1917, p. 70), Eddington (1920, p. 97) or Lodge (1921, p. 555) did use the metric for all $r$ > 0. Putative crossing under the $r=2m$ “singularity” was also discussed at a 1922 Paris seminar of Einstein, as reported by Nordmann (1922, pp. 154-156) and Brillouin (1923).
The possibility was dismissed then, and apparently never seriously considered again until the early 1960s, when people discussed extended metrics.
(Eisenstaedt also notes that Weyl had pioneered such an extension in (1917, p. 131; 1918, p. 205), but removed it from later editions of his book.)