Depending on how sure I am of where I'll be fishing and what I'm fishing for, as a banky I may have separate rods for bobber fishing, plunking, and drifting/hardware. From the kayak I'll have hardware, trolling, jigging, and bottom fishing (something carolina rigged). If I'm going something very specific like surf perch, I'll have just the one, with a spare in the truck in case of equipment failure.
Probably the biggest thing I've learned around this is to not bring a full tackle box. I bring enough to re-rig everything I've brought 3 times and that's it. If I lose my rig twice quickly, I'm doing the wrong thing in the wrong area, so I'm shifting techniques and/or moving. I only bring what fits in my pockets/vest, or if I know exactly where I'll be, I'll bring a little more, in a small ammo box from Harbor Freight.
I leave extra terminal tackle back at the truck, but still keep the quantity within reason. I picked up one of those new flat ammo boxes from BiMart which fits nicely behind the seat in the truck and is easily packed for the day's trip, whether that be fishing, hunting etc. Very easy to pack in advance and just grab it and go.
Edit: I misinterpreted the question - to the river, maybe 2 as I already know where I am and what approach I'm taking. The others stay in the truck. Also I answered this as generic, not steelhead specific.