It seems to me that tradition may have overlooked an omni-attribute of God, namely God's omni-connectedness. If God connects to an entity, then God establishes a link or bridge between himself and the entity in question. That is to say, God relates to it or associates himself with it, which goes beyond God knowing it, sustaining it, or being closely present to it. God being omni-connected thus means that God binds himself to every other entity.
One may respond that omni-connectedness is already implied by God's other omni-attributes, such as omni-presence, omni-benevolence and omni-potence. Yet, it seems to me that these other omni-attributes are prerequisites for being omni-connected or qualifications of the way in which God is omni-connected. But then they are not sufficient for omni-connectedness to obtain. For that it is still required to posit omni-connectedness as an additional omni-attribute of God.
I would treat omni-connectedness similar to God's other omni-attributes. As part of God's essence omni-connectedness is a necessary property of God. One may worry that God being omni-connected entails God being connected to evil. But this would only follow if evil is itself an entity, which is implausible. Besides, even if evil would be an entity, to be connected to it does not entail to endorse it or to be responsible for it.