Pretty cool to have a cow tag in the Umtanum and Manastash; you shouldn't have any trouble finding animals. The spikes are ghosts during the general season, but even with all the traffic you could probably find a cow without too much trouble.
Before the snow, it seems like elk are west in the timber still. Whereas the lowland and sagebrush areas are to the East toward the river. Looks like 340 has more timber than 342.
You're on the right track here. I saw a few cow elk in the eastern sections opening weekend of deer, but they're very few and far between. Most of them will still be in the timber this week enjoying the last of the green grasses and relative shelter. Although it's always hard to tell what a little rain and wet snow will do.
Personally, I'd be focused on the transition zones after the general season ends. Good luck!