I think you are confusing the soldier that was initially infected to the one that rises from the "dead" to attack Richard. The soldier that was initially infected is this guy who touched Himiko directly with his bare skin. Note you can see the third Trinity soldier in the background of this image:
However, the Trinity soldier that rises from the "dead" to attack Richard is actually this guy.
As you can see he is a completely different guy, one who was not actually infected when he "died". He "dies" when he is shot by the aforementioned infected soldier, though as you can see he is wearing a bulletproof vest so he presumably just got knocked unconscious.
Given this it actually remains unclear how this soldier got infected. He might not actually have been infected but given his actions and that Richard then apparently gets infected after the fight I would say he must have been.