Essentially, yes. An old result of Kleene [1], later strengthened by Craig and Vaught [2], shows that every recursively axiomatizable theory in first-order logic without identity, and every recursively axiomatizable theory in first-order logic with identity that has only infinite models, has a finitely axiomatized conservative extension. See also Mihály Makkai’s review, and Richard Zach’s summary.
References:
[1] Stephen Cole Kleene: Finite axiomatizability of theories in the predicate calculus using additional predicate symbols, in: Two papers on the predicate calculus. Memoirs of the American Mathematical Society, no. 10, Providence, 1952 (reprinted 1967), pp. 27–68.
[2] William Craig and Robert L. Vaught: Finite axiomatizability using additional predicates, Journal of Symbolic Logic 23 (1958), no. 3, pp. 289–308.