Some bushwalking ways end at the river side, implying that the route crosses over and starts again directly on the other side.
Before crossing, double check that the way indeed crosses over, rather than continuing along the same river bank. Do this by having a quick scout around on the side of the bank you’re on, and looking over to the other side. Sometimes, there’s a clear sign where the way starts again, or there’s a signpost, a small arrow, or a marker. Don’t be surprised though if the way only becomes visible once you’re on the other side: many river crossings are implied rather than signposted.
Assess if the river is safe to cross. Then choose where to cross the river, bearing in mind that this might involve detouring slightly upstream or downstream of the path. Then continue along the way again.
If you cannot pick up the way again after the river crossing, cross back to the way you were on and go back to the last known point. You may have accidentally strayed from the main way, or picked up a way leading off the main one to the river (typical for lunch or swimming spots).
Maps are not that reliable when it comes to river crossings. Ways change over time, and where the way originally crossed the river, it may not do so any more.
The best general advice is to follow the main way whenever given a choice. That is, if the main way continues on one side of the river and you spot another smaller way on the other side of the river, then don’t cross over, stick to the main one. Conversely, if the main way continues on the other side of the river, and only a minor one follows on the same side, cross over.