Hi Noah, I'm adding an answer because this wouldn't fit in a comment. Unless I'm making a silly mistake, the situation is different with respect to adding an escaping real:
Let $P$ be the collection of finite partial functions from $\omega$ to $\omega_1$ as usual, and let $\dot f$ be a $P$-name for the generic surjection from $\omega$ onto $\omega_1$.
Given $g:\omega\rightarrow\omega$ in the ground model, and $n<\omega$, consder the set $D(g, n)$ of conditions $p$ such that for some $m>n$,
- $m$ is in the range of $p$,
- the domain of $p$ is an initial segment of $\omega$, and
- the least $k$ for which $p(k)=m$ is greater than $g(m)$.
This set is dense in $P$ for each $g$ and $n$, and so the real $h$ in the extension defined by setting $h(m)$ equal to the least $k$ such that $\dot f(k)=m$ is not bounded by a ground model real. (So essentially, we are "inverting" the surjection on the initial segment $\omega$ of its range)
Edit: Even simpler, if we define a real $h$ in the extension by setting $h(n)=\dot f(n)$ if $\dot f(n)<\omega$, and $h(n)=0$ otherwise, then $h$ is Cohen over the ground model, hence unbounded.