There's really not much written about Hood's and Whiting's attacks at Fair Oaks Station on May 31st. Since the area was held by Federal troops after the fighting ended, Confederate dead would not have been identified or formally buried with anything but the simplest of temporary markers.

Confederates referred to the battle of May 31/June 1, 1862, as Seven Pines, where Southerners broke Federal lines but suffered heavy losses. If the place of death was recorded on company rolls as Seven Pines, it probably means that the soldier was killed or mortally wounded in action during this battle.

Other than the action at Eltham's Landing on May 7th, Hood's Brigade wasn't involved in any fighting until May 31st. In 1864 Grant and Sherman made a deliberate effort to keep their armies engaged on an almost daily basis, meaning that infantry commands skirmished frequently. In 1862, the two armies tended to maintain a respectful distance, leaving most scouting and skirmishing to the cavalry. If this man died of wounds in May/June 1862, his injuries were likely suffered in a major engagement.

I'd suggest that you contact the National Park Service and ask about mass graves in the Fair Oaks area.