Northwest River Forecast Center
If you click on the point at the dam it will give a projected inflow in cfs, in a menu on the far right.
This gauge rarely tracks Rome and the projections are rarely accurate. But it can be useful for looking at trends. I've spoken to them about this, the inflow is calculated by measuring the rise in the reservoir, not the most accurate method. No excuse was given for why the predictions are so poor.
Remember to carry firewood, regs require it and frankly there isn't much down there.