Do very short values make a peppered hash less secure?
The assumption seems to be that the adversary wants to confirm a guess of $\mathtt{value}$ given $\mathtt{hash}=\operatorname{SHA-256}(\mathtt{value}\mathbin\|\mathtt{pepper})$, for unknown random secret $\mathtt{pepper}$. This is an ad-hoc PRF of $\mathtt{value}$ with symmetric key $\mathtt{pepper}$.

No, the small size/entropy in $\mathtt{value}$ is not an issue. Neither is having a million of millions of $\mathtt{value}$ (or/and of leaked $\mathtt{hash}$ ). Without a successful and complete guess of $\mathtt{pepper}$, as far as we know, the adversary can't learn anything about $\mathtt{value}$ from its $\mathtt{hash}$ (beyond two $\mathtt{value}$ being identical with practical certainty when their $\mathtt{hash}$ is, something that additional random public salt would fix, see Maarten Bodewes's answer). Main theoretical weakness is the relatively small size of $\mathtt{pepper}$ (here 80-bit), which should be 128-bit or more by modern standards. Of course, there's the issue of keeping $\mathtt{pepper}$ secret, and other implementation issues (but the classic timing side channel of memcmp does not matter).

The best known generic attack is essentially brute force. For each guess of $\mathtt{value}$, it enumerates $\mathtt{pepper}$, computes $\operatorname{SHA-256}(\mathtt{value}\mathbin\|\mathtt{pepper})$, and compares to the given $\mathtt{hash}$ (or the full list of theses). If there are several guesses of $\mathtt{value}$ with different likelihood, it pays to try the guesses from most to least likely. If $\mathtt{value}$ was long (so that $\mathtt{value}\mathbin\|\mathtt{pepper}$ exceeds the 64-byte block size of the SHA-256 internals), it would pay to cache the intermediate values that do not vary with each $\mathtt{pepper}$.

Note: we'd have better academic security assurance if we used HMAC-SHA-256 rather than this ad-hoc PRF. See M. Bellare: New Proofs for NMAC and HMAC: Security without Collision-Resistance, in Journal or Cryptology, 2015, originally in proceedings of Crypto 2006.