White House became the AoP's supply base until Grant crossed the James. Grant seemed to work well with the Navy; for whatever reason, Butler had problems at times. After Cold Harbor, Baldy Smith's Corps was transported back up the James for the assault on Petersburg. There did not seem to be any problem in either direction.Grant actually did make a move like this during the Overland Campaign -- it just wasn't on the James River towards Petersburg.
Smith's XVIII Corps (16,000) was detached from Butler, moved down the James River and up the York River to the Pamunkey, arriving at White House on May 30-31. A brigade was left to safeguard White House and Smith moved with about 10,000 to join Grant/Meade's left flank at Old Cold Harbor on June 1 (moving mid-afternoon).
Grant sent Sheridan ahead to secure the crossroads and pulled Wright's Corps from the right, moving behind the Army, to extend and reinforce the left. The idea was to push through Old Cold Harbor toward Richmond. This would put Grant's army around Lee's right flank, south of Lee and closer to Richmond, with an open road to the Confederate capital and Sheridan's cavalry massed to exploit the breakthrough.
This could have worked, maybe should have worked. Great effort and initiative by Robert E. Lee, combined by some fierce Confederate resistance and missed Union opportunities, barely saved the day. That is how we got to what history calls the Battle of Cold Harbor, a bloody repulse if ever there was one.
As an aside, the Confederates had the means to mine the shipping channel in the Pamunkey, but did not seem to have the will.