Others can flesh this out and/or correct anything wrong, but calling your elite desk ahead of time (starting right after your upgrade window closes) might get you a seat that inventory control released if it's before the automatic process runs again (I think that only runs 2 or 3 times a day). For example, the Silver window closes at 2 days prior. If no automatic upgrade clears, calling an hour later may snag a seat that inventory control just released for upgrade/award even though there may be Chairman, Platinum, and Gold that also didn't get the auto upgrade and are ahead of the Silver in priority.
As far as checking with the GA, that is just to make sure that your name is on the upgrade list at the airport and that you didn't fall through the cracks somewhere.
Jim