Direct proof of a strongly $\lambda$-homogeneous elementary extension A structure $M$ is strongly $\lambda$-homogeneous if a partial elementary map $A \to M$ for $A \subseteq M$ of cardinality less than $\lambda$ extends to an automorphism of $M$.  For a fixed cardinal $\lambda$, every structure has an elementary extension that is strongly $\lambda$-homogeneous.
The proof of the existence of strongly $\lambda$-homogeneous elementary extensions goes by way of $\lambda$-bigness in the sense of (longer) Hodges.  Here, a structure $M$  is $\lambda$-big if $(M, \bar a)$ is resplendent for any tuple $\bar a \in M$ of length less than $\lambda$, and a structure $N$ is resplendent if for any theory $T$ in an expanded language consistent with that of $N$, some expansion of $N$ models $T$.  That every structure has an elementary extension that is $\lambda$-big can be proved easily, but the proof requires careful bookkeeping and counting.
Can the existence of strongly $\lambda$-homogeneous elementary extensions be proved more directly and elementarily (from ZFC)?  In case it matters, I'm interested in the case where $\lambda = \aleph_0$.
 A: The other standard way to get strongly $\kappa$-homogeneous elementary extensions in ZFC is by way of the theory of special models. A special model $A$ of cardinality $\lambda$ is one that can be written as a union $A = \bigcup_{\kappa<\lambda}A_\kappa$ of an elementary chain indexed by the cardinals less than $\lambda$, such that each $A_\kappa$ is $\kappa^+$-saturated.
This is also explained in Hodges's Model Theory, Section 10.4. The relevant results are Theorem 10.4.2(b):

Let $A$ be an infinite $L$-structure and $\lambda$ a strong limit number $>|A|+|L|$. Then $A$ has a special elementary extension of cardinality $\lambda$.

...and Corollary 10.4.6. Unfortunately, there's a typo here (which is apparent from reading the proof). Corollary 10.4.6 should read:

If $A$ is special of cardinality $\lambda$, then $A$ is strongly $\text{cf}(\lambda)$-homogeneous.

So to get a strongly $\kappa$-homogeneous elementary extension of a model $A$, you find a strong limit cardinal $\lambda>|A|+|L|$ with $\text{cf}(\lambda)\geq \kappa$, for example: $\lambda = \beth_{\kappa^+}(|A|+|L|)$, and build a special model of cardinality $\lambda$.
You can also read about special models in Section 6.1 of A Course in Model Theory by Tent and Ziegler, where the relevant results are Corollary 6.1.3 and Theorem 6.1.6.
The construction of special models is actually fairly elementary - it's not too much harder than the construction of $\kappa$-saturated models. But maybe you don't find it "direct". It might be possible to construct strongly $\kappa$-homogeneous models by a direct transfinite induction argument - in fact I remember working out such an argument many years ago (of course, I'm not at all sure now that it was correct). But any such argument will almost certainly involve much more bookkeeping than the argument via special models. The idea was to add the new automorphisms to the language as function symbols, to make sure their existence is preserved under extensions. But then one has to think about extensions which are elementary in the original language, which also satisfy axioms about the function symbols in the expanded language... it's a bit of a mess.
