If $C$ has an action of $\mathbb Z$, then $C\otimes C$ has an action of the wreath product ${\mathbb Z}\wr {\mathbb Z}/2$. This is the split group extension $1\to {\mathbb Z}\times{\mathbb Z}\to  {\mathbb Z}\wr {\mathbb Z}/2\to {\mathbb Z}/2\to 1$ associated with the natural action of ${\mathbb Z}/2$ on ${\mathbb Z}\times {\mathbb Z}$. Inside this group you have the subgroup ${\mathbb Z}\times {\mathbb Z}/2$, where ${\mathbb Z}$ is the diagonal subgroup of ${\mathbb Z}\times {\mathbb Z}$.

With this notation, $h(hSym^2C)/{\mathbb Z}$ is the homotopy quotient of $C\otimes C$ by ${\mathbb Z}\times {\mathbb Z}/2$, while $hSym^2(hC/{\mathbb Z})$ is the homotopy quotient of $C\otimes C$ by ${\mathbb Z}\wr {\mathbb Z}/2$. So your map can be understood as the quotient map 
$$[C\otimes C\otimes E({\mathbb Z}\times {\mathbb Z}/2)]_{{\mathbb Z}\times {\mathbb Z}/2}\to [C\otimes C\otimes E({\mathbb Z}\wr {\mathbb Z}/2)]_{{\mathbb Z}\wr {\mathbb Z}/2}$$
associated with the group inclusion ${\mathbb Z}\times {\mathbb Z}/2\hookrightarrow {\mathbb Z}\wr {\mathbb Z}/2$. In particular, the map is not an isomorphism, or even a chain homotopy equivalence, except in the most trivial case.

You can make this map pretty explicit by choosing nice chain level models for $E\mathbb Z$, $E{\mathbb Z}/2$ and $E{\mathbb Z}\wr {\mathbb Z}/2$. As you say, a good model for $E{\mathbb Z}/2$ is given by cellular chains on $S^{\infty}$, with the standard cell structure that has two cells in each dimension. To get a nice model for $E{\mathbb Z}$, think of $\mathbb Z$ acting on $\mathbb R$ by translation. Equip $\mathbb R$ with a cell structure with a zero cell for every integer $n$, and a $1$-cell for every interval $[n, n+1]$. The cellular chain complex gives you a nice small model for $E\mathbb Z$. Explicitly, it has the form ${\mathbb Z}[{\mathbb Z}]\leftarrow {\mathbb Z}[{\mathbb Z}]\leftarrow 0 \cdots $, where the boundary homomorphism is determined by $[n, n+1]\to [n+1]-[n]$. Once you have models for $E{\mathbb Z}$ and $E{\mathbb Z}/2$, the chain complex $E{\mathbb Z}\otimes E{\mathbb Z}\otimes E{\mathbb Z}/2$ gives a model for ${\mathbb Z}\wr {\mathbb Z}/2$.