As Asaf and Joel have observed, the answer to your question is negative. However, there is a sense in which being an elementary submodel of $L_{\omega_1}$ is the only way to "persistently" get elementary submodelhood relations.

Specifically, the following are equivalent:

 1. $L_\alpha\prec L_{\omega_1}$.

 2. There is a club $S\subseteq\omega_1$ such that $L_\alpha\prec L_\beta$ for all $\beta\in S$.

 3. There is an unbounded set $X\subseteq \omega_1$ such that $L_\alpha\prec L_\beta$ for all $\beta\in S$.

In some sense, if $L_\alpha\not\prec L_{\omega_1}$, then there is a limit to how many times we can elementarily extend $L_\alpha$ within the $L$-hierarchy. (But you have to be careful here, since we *can* have an infinite ascending $\prec$-chain of countable $L$-levels not elementarily embedded into $L_{\omega_1}$.)

The implication from 1 to 2 is via dLS+Condensation directly (just build the club) together with the **coherence** property of $\prec$, and the implication from 2 to 3 is trivial since clubs by definition are unbounded.  The direction 3 to 1 is a bit trickier. Note that we **cannot** directly use elementary chains, since given $X$ appropriate there is no guarantee that $L_\beta\prec L_\gamma$ for $\beta<\gamma$ elements of $X$. Instead we have to use definable Skolem functions, which each level of the $L$ hierarchy has (uniformly, even).

****

But on the *other* other hand, if $V=L$ then there **is** an unbounded $U\subseteq \omega_1$ such that for all $\alpha,\beta\in U$ we have $L_\alpha\equiv L_\beta\not\equiv L_{\omega_1}$ *(note that this can't be proved using just a counting argument!)*. This is a beautiful short application of Tarski's undefinability theorem due to Hjorth, answering [question 10.4](https://people.math.wisc.edu/~awmille1/res/problems.pdf) of A. Miller. Hjorth's [argument](https://people.math.wisc.edu/~awmille1/res/unpub/hjorth-10.4.txt), with minor formatting edits from me, is copied below (which I've left hidden to avoid spoilers):

>! Let $X$ be the set of complete theories that satisfy "everything
is countable" and have unboundedly many $\alpha<\omega_1^L$ 
with $L_\alpha$ realising them. The theory of $L_{\omega_1^L}$ is one such theory, and we will be done if we prove that there are some others. Now $X$ is a definable class in $L_{\omega_1^L}$, and so it must have some other elements or else $L_{\omega_1^L}$ would admit a truth defintion ($\varphi$ is true in $L_{\omega_1^L}$ iff the unique element of $X$ contains $\varphi$).