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[SoCG'24] obtained Steiner $(1+\\varepsilon)$-spanners of size $O_d(\\varepsilon^{(1-d)/2}\\log(1/\\varepsilon)n)$, nearly matching the lower bounds of Bhore and T\\'oth [SIDMA'22].\n  We obtain Steiner $(1+\\varepsilon)$-spanners of size $O_d(\\varepsilon^{(1-d)/2}\\log(1/\\varepsilon)n)$ not only in $d$-dimensional Euclidean space, but also in $d$-dimensional spherical and hyperbolic space. 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