The Three-Day Hunt by Robert R. Chase (Asimov’s Science Fiction, January/February 2021) starts with an Afghanistan veteran called Hammond going to investigate a flying saucer that has crashed near to his cabin in the woods. When he and Tripod, his three-legged dog, get to the craft the pilot is missing, so they start tracking it.
The rest of the story has the pair following the alien through the wood for the next couple of days, during which we get Hammond’s military and domestic backstory as well as the dog’s (their paths crossed in Afghanistan, just before a bomb went off and injured them both). Later, the military contact him by phone to try to get him to stop his pursuit, but Hammond ignores them and carries on.
Then (spoiler), when Hammond stops to treat the dog’s bleeding paws, he finally sees the alien. As Hammond approaches it, the alien gestures towards the dog—at which point the story dissolves into a mini-lecture about how humanity’s domestication and/or symbiosis with dogs makes it more likely that we will be able to successfully establish a relationship with aliens.
More a notion than a story, but okay, I suppose.
** (Average). 4900 words.