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](https://doi.org/10.2307/2270294), and Richard Zach’s [summary](http://richardzach.org/2008/05/14/finite-axiomatizability-of-theories-in-the-predicate-calculus-using-additional-predicate-symbols-classic-logic-papers-pt-4/).

**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.