An early occurence of such bounds is in the theorem of Theorem of vonBahr and Eseen
vonBahr, B., Esseen C.-G.: Inequalities for the rth absolute moment of a sum of random variables, $1\leq r \leq 2$. Ann. Math. Statist. 36, No.1, 299-393 (1965).
Theorem: Let $X_i$ be independent (not necessarily i.i.d.) zero mean random variables. Then, for $\alpha\in [1,2]$, $$E|\sum_{i=1}^n X_i|^\alpha\leq C_\alpha \sum_{i=1}^n E|X_i|^\alpha\,.$$ ($C_\alpha$ is explicit)
Applying Markov's inequality in your setup gives $$P(|S_n|>t)\leq C_\alpha n E|X_1|^\alpha t^{-\alpha} $$ as you wanted.
Note: the moment condition certainly does not imply convergence to $\alpha$-stable - one would need some regularly varying conditions on the tail for that.
Note2: A good summary (up to late 70s) of estimates of this kind (mostly from the Russian school) are in Nagaev's Annals of Probability paper (1979). Petrov's 1975 book is also a good resurce - in particular the VonBahr-Essen bound is mentioned there (on page 60).