by the few turns it takes to get a warrior to the end of it, there's practically nowhere left for me to expand into.
If you're stuck on a peninsula as a Seafaring-Civ, especially on an 80% water map, you might be better off building Curraghs for exploration.
Seafarers start with Alph, so you can start doing that straight away at Monarch (where 2 citizens are born content) because you won't get riots until Pop3. Conversely, at Emperor+ (where only 1 citizen is born content) you will probably need to build a Warrior first for MP at Pop2.
Although a Curragh will take a bit longer to build (8T at the 'standard' 2 SPT, instead of 5T for a Warrior), a
Seafaring Curragh's M=3 will make up that 'lost' 3T within its first 2 moves (moving 6 tiles, when a Warrior, though finished more quickly, would still only have moved 5 tiles). And if a Curragh is your second build (after the MP-Warrior), then you might even be able to get an extra 1-2 SPT at Pop2 which could shave 1-2T off the remaining build-time.
And if you send out 2 Curraghs (30 shields) from your capital to map your own continent/island's coastline (one going north/west, one south/east), once they've met up at the other end, then they can also start looking for potential routes to visit the neighbouring islands/continents — which an exploring Warrior(s) obviously can't do!
Of course, you should certainly also send out your first Settler ASAP — but since you wouldn't be able to complete it until Pop3 at the earliest, then it could be your third build (10T at 3SPT at Pop2-3) after the Warrior + Curragh. Once the first Settler is on its way, your first Warrior can go with it — or explore further inland (depending on landform) — while your capital starts over with another Warrior + Curragh (+ Settler?) sequence.