I don't know if this is actually true, or is a superstition on my part, but I usually search first for Saver Awards, which gives me some idea of what flights are more likely to have capacity, then search for Upgrade Awards on days where there are Saver Awards. In other words, I am (perhaps falsely) intuiting a link between "I" fares and "R" fares.

Also, while the advice to start your searches from the hub is correct, you should also search end-to-end. For reasons I don't understand, sometimes end-to-end upgrades appear when hub-destination ones do not. This has proven true for me on multiple occasions.

Finally, consider searching for one-way upgrades first, and back-forming your r/t from those.

Good luck. Maybe someone with expertise on Expert Flyer can tell us if there is an easier way -- I'm just learning that system.