"...something else after flushing would have been good to use?..."
Nope, nothing other than water. Just plain cool running water for at least 10 minutes. Then get thee to a doctor or emergency room immediately. Medical attention is especially important for a chemical exposure to the eyes, nose, mouth, or the face in general.
In case people are wondering about the common wisdom to use vinegar for a lye splash, the answer is NO, NEVER. In a strict chemistry sense, yes, vinegar does neutralize lye. But the neutralization reaction releases heat which damages tissues, and the vinegar itself can do its own damage since it is an acid.
So if you use vinegar on a lye splash on your body, you will get a thermal burn AND a caustic burn AND additional pain and inflammation from the vinegar itself. The goal should be to wash the darn stuff off with simple water as quickly as possible.
Plenty of cool plain running water is the standard first aid in chemical laboratories and manufacturing plants for a chemical splash. It should be the soap maker's first aid method too.