We're sort of delving into next weeks discussion but early in part 4 of the same chapter -

"The mysterious character of the commodity-form consists therefore simply in the fact that the commodity reflects the social characteristics of men's own labour as objective characteristics of the products of labour themselves, as the socio-natural properties of these things." pg 164-165

Marx is also saying that there is this "something else", something mystical in the commodity that its natural-form, as a product of human concrete labor, cannot be the source of. Instead, it is the commodity-form itself that is the source of this mystical property. And in this I think he means the social relations between humans that must exist for commodities to be produced in the first place. These social relationships become objectified in the products of human labor; in our alienation with the products of our own labor, in the fact that we produce solely for exchange, and in the fact that we are humans with human consciousness that means we have desires, dreams, and values of our own that may conflict with commodity production.

In other words, it is this social aspect, this territory of relations, that qualifies the labor marx is talking about here. Human labor (I would think consciousness fits here in our analysis, or at least a particular degree of consciousness), social labor (production for exchange), private labor (alienated from the products of our labor. Someone else owns it and the means of production).

So I think this feeling of "there must be something else beyond just labor" is pointing to the context that qualifies this labor as something very particular. But this context is a set of relationships that are difficult if not meaningless to examine outside of their broader context of "being in relation" to something. So when Marx focuses on the distinction between concrete and abstract labor in the first part of chapter 1, we're still left thinking "but...isn't there more?".