Just an idea, but as far as I know there is no reason that space cannot expand faster than the speed of light. If the big bang created space or if an equivalent event in existing space emerged as a ball of energy without mass, then the superluminal expansion in the first picoseconds could have occurred. After that inflationary epoch, matter could have formed from the energy density.

In this scenario matter would not have to travel faster than the speed of light, and yet would still be evenly distributed across the entire expanding ball after it forms.