Does every r.e. set containing the set of total recursive functions contain all partial recursive functions? Any r.e. subset of $A\subseteq\mathbb{N}$ which contains the set $$\mathrm{Tot}=\{i\mid i\ \mbox{is an index of a total function } f\}$$ must, by a standard argument (of Post?) contain some partial recursive function indices.

Given a partial function index (and every total function), it's pretty easy to construct many others, e.g. an index for the function which returns $0$ on prime inputs and is undefined otherwise.

But must $A$ contain all partial functions? This seems like a simple question, but I can't find an argument one way or the other.

Edit: I'm equally (more, actually) interested in the case where $A$ is recursive, e.g., represents the programs from some programing language.
Take $A = \{i\ |\ \phi_i(0) \downarrow \}$. It is obviously r.e., and includes $\mathrm{Tot}$.

Yet, it does not contain any index of, say, the always-undefined function $u$. Nor indices for any recursive partial function which is undefined at $0$.

The Kleene set also works: $K = \{i\ |\ \phi_i(i) \downarrow \} \supset \mathrm{Tot}$, yet it does not contain any index for $u$. (And it is r.e.)