[Adm.Lee]

They should be issuing orders to units that still exist, or working hard to make sure they stay that way! Someone needs to be looking at the bigger picture, above the division commanders, after all.
For instance, the US Sixth Army in 2000 seems to be the controlling HQ for the California side of the Mexican front, while Fifth Army covers the Texas-Oklahoma side. Both of those should be seeking supplies in their rear areas and funneling them to the front forces.
Given the breakdowns in communications, those generals and their staffs should be talking to governors and mayors, and they are effectively representing MilGov to those authorities. Where can they help, how can they obtain stuff?

The way I see things, when the cantonment system arose, Army and Corps HQs would be coordinating things among their subordinate formations. Most of the Cold War corps should have some rear-area defense/theater-defense brigades assigned in addition to their small MP units, so those would be patrolling roads and escorting convoys.
Engineer brigades are assigned to corps and armies, so those would be working at defenses and roads.
Most heavy artillery is at the corps level when the war starts, those might be moved into central firebases and used to cover the front lines.
Signal brigades may have developed pigeon and courier-rider systems, too.

IMC, I think that grounded air force units would be: a) cannibalized for personnel to fill up Army units, and b) reorganized into more road-security and farm-security units in the rear areas. These would be controlled by the corps and/or army HQs.

My understanding of the US Army, at least, is that Corps HQs were meant to stay out of logistical measures as much as possible (not exclusively), focusing on tactical concerns, while Army HQs were the other way around. Other armies probably do it differently?

Where Corps and Army HQs are is something I've devoted (wasted?) too much time on, after all.