Generate a new planar graph via edge subdivision and edge addition

Question

Given a simple planar graph $G$, let $G'$ be a subdivision of $G$. Examples of $G$ and $G'$ are shown below:

The graph $G'$ is clearly planar. Let $W = \{w_{i,j} : (v_i, v_j) \in E(G)\}$ be the set of newly added vertices in $G'$. Consider the following procedure of creating another graph $G''$:

1. Label all vertices in $W$ unvisited.
2. Start from any $w \in W$ as a source node. Label $w$ as visited.
3. Add a new edge connecting the source node to an unvisited $w_{i,j} \in W$ in $G'$.
4. Label $w_{i, j}$ as visited.
5. Let $w_{i,j}$ be the new source.
6. Repeat steps 3,4, and 5 until all vertices in $W$ are visited.

Here is an example of $G''$ after the above procedure (newly added edges are highlighted in blue):

Clearly, there are many ways to generate different $G''$ via the above procedure. The goal, however, is to make $G''$ also planar. Thus, the question is, how to select the next unvisited vertices in step 3 such that the resulting graph $G''$ is planar?

My intuition is that there always exists a way to construct such a planar $G''$. I will keep updating this question if I have new thoughts.