It can be assumed that Agent 4 has canonly worked Salmon Run shifts before, as there are several lines of written dialogue from Callie and Marie about Salmon Run that can be used if one has played the mode. Agent 8 has also probably worked some shifts, as Salmon Run lobbies (at least mine) are usually majority octoling, and it would be cool if Nintendo recognized that. Additionally, DJ Octavio is in charge of the Octarians (or at least he used to be, ha ha!) who would trade with the Salmonids. So, what if there's a plot where the New Squidbeak Splatoon is in support of Grizzco because Four and Eight were brainwashed into following them and the big kids are too busy with other stuff to notice, meanwhile DJ Octavio wants to stop Grizzco because they steal eggs and prevent the few soldiers he has left from being able to trade with the Salmonids, so you team up with him.
I wouldn't really like that plot, but I can see Nintendo doing it.
A more interesting one would be a story where the eggs were used to build an entire robot army and the player (Agent 5?) travels across the five islands of the Great Salmonid Migration (the Salmon Run stages) to help out locals by defeating the robots and completing tasks for them. Completing tasks gives you some cash that you can eventually use to buy a boat ticket to the next island, but each island has a different currency. All the while, you're trying to investigate the mysteries Grizzco. Maybe Agent 4 can help you out every now and then? This is just an idea, but it seems cool in my head...