hmac won't fall back if OpenSSL is available, the requested algorithm isn't in OpenSSL, but the algorithm is in hashlib.
If you [monkey]patch hashlib to include a new algorithm, you can't use that algorithm from hmac by name.
It appears that the OpenSSL implementation (known as _hashlib from inside hashlib, or _hashopenssl from inside hmac) doesn't actually return an UnsupportedDigestmodError, but rather it's base class ValueError.

MRE

# The following is MRE-specific to easily introduce a new name
# My use case involves a monkeypatch, but imagine any algorithm NOT implemented by OpenSSL, ONLY by hashlib
>>> hashlib.__builtin_constructor_cache['myhashalg'] = hashlib.md5
>>> hashlib.new('myhashalg', b'').digest().hex() # confirm hashlib can use that name
'd41d8cd98f00b204e9800998ecf8427e'
>>> hmac.digest(b'key', b'message', 'myhashalg')
Traceback (most recent call last):
  File "<pyshell#nnn>", line 1, in <module>
    hmac.digest(b'key', b'message', 'myhashalg')
  File "C:\Python311\Lib\hmac.py", line 198, in digest
    return _hashopenssl.hmac_digest(key, msg, digest)
ValueError: unsupported hash type myhashalg

The exception goes unhandled at

This also shows up in the stateful (non-oneshot) code at

Environment

Windows 10 64-bit
Python 3.11.2

Possible fixes

• Change _hashopenssl.hmac_digest to correctly raise an UnsupportedDigestmodError (this looks like what was intended, given bpo-40645: use C implementation of HMAC #24920)
• Catch a ValueError instead (as UnsupportedDigestmodError is derived from ValueError this would work, but may not be what is truly intended)
• Something closer to what existed before like https://github.com/tiran/cpython/blob/837f9e42e3a1ad03b340661afe85e67d2719334f/Lib/hmac.py#L181 ??

Linked PRs

• gh-103256: Fix hmac algorithm to support fallback implementation #103286
• [3.11] gh-103256: Fix hmac algorithm to support fallback implementation (gh-103286) #103328