Constructions are easier in the uniform setting.
When doing reductions to SAT express the problem
you want to reduce to SAT as a SO$\exists$ formula
(by Fagin's theorem in descriptive complexity
SO$\exists$ captures all queries computable in NP).
Then convert it into a propositional formula using propositional translation.
Then convert that to a CNF using Tseytin transformation:
Let the input $G$ be given by the number of vertices $n$ and
the edge relation $E$.
To simplify things let's assume that each vertex has an edge to itself,
i.e. $E(v,v)$ for all $v$.
Let $S$ be a unary predicate that determines the set of selected vertices.
SO$\exists$ formula for your problem: $\varphi(G,S)$ is
$$\forall s,t \in S \ \exists P
\text{ ``$P$ is a path from $s$ to $t$ in $G[S]$''}$$
To express that $P$ is a path from $v$ to $u$ in $G[S]$
we think of $P$ as an ordered list of size $n$ of vertices,
i.e. $P$ is a function from $[n]$ to set of selected vertices:
$$\forall i \in [n] \ \exists v \in S \ P(i,v) \land
\forall i \in [n] \forall v\neq u \in [n] (\lnot P(i,v) \lor \lnot P(i,u))$$
and it is a path from $s$ to $t$:
$$P(1,s) \land P(n,t) \land
\forall i \in [n-1] \ \forall v,u \left(
P(i,v) \land P(i+1,u) \to E(v,u)
\right)$$
Now we turn it into a propositional formula using
the propositional translation.
We use $q_v$ to express vertex $v$ is in $S$, i.e. $v \in S$.
We use $p_{iv}$ to express $P(i,v)$.
Note that when translating
we need to use different variables for expressing
the path between each pair.
It is really $p_{iv}^{st}$.
$$\bigwedge_{s,t \in [n]} \left(q_s \land q_t \to ... \right)$$
where $...$ is the conjunction of
$$\bigwedge_{i\in [n]} \bigvee_{v \in [n]} (q_v \land p_{iv}^{st}) \land
\bigwedge_{i\in[n]} \bigwedge_{v\neq u \in [n]}
(\lnot p_{iv}^{st} \lor \lnot p_{iu}^{st})$$
and
$$p_{1s}^{st} \land p_{nt}^{st} \land\bigwedge_{i \in [n-1]} \bigwedge_{v,u \in [n]}\left(
p_{iv}^{st} \land p_{i+1u}^{st} \to e_{vu}
\right)$$
This has polynomial size in $n$.
To make it CNF perform the reduction from SAT to CNF-SAT using
Tseytin transformation:
use new variables (called extension variables) for each subformula
and expressing the relationship between the variable for each subformula
and the variables for its intimidate children in the formula tree and
that the variable for the root subformula is true.
The result is a polynomial-size CNF formula $\psi(\vec{e}, \vec{q}, \vec{p}, \vec{z})$ with variables $\vec{e}$, $\vec{q}$, $\vec{p}$,
and a bunch of extension variables $\vec{z}$ (for subformulas).
Fix $\vec{e}$ based on the given $G$.
Any assignment $\tau$ to $\vec{q}$, $\vec{p}$, $\vec{z}$ that satisfies $\psi$
gives a connected subgraph $S = \{v \mid \tau(q_v) = \top \}$ of $G$ and
for any connected subgraph of $G$ there is such an assignment.
