I used to carry a small syringe, but now I use Steripods or one of its imitators. I am not sure if they match what you used. They are single-use with a tab you break off. In my experience the nossle hole left is small enough for squirting - you aren't left dripping the fluid in. In fact I just tested, and could send a stream of fluid 10 or 15 feet. I hope that's enough for irrigation.

Each holds about 20ml. The more you have the better, of course, but just one is better than nothing as you can refill them with clean water for more squirting.

I sometimes also carry an antiseptic (TCP) but I'd hesitate before using it as real medics seem to disapprove. (I have no medical training.)