How to sequence SHA256 block circuit to obtain hash of a message comprising multiple 512 blocks?
One can't use the circuit exactly as given.

To build a two-blocks SHA-256, the second round must be fed as input both the second 512-bit (padded) block, and the 256-bit output by the first round. The final hash is then the 256-bit output of the second round.

For the first round, and in the circuit given, the only variable inputs are the 512-bit first block; the other 256 input bits are always set to a constant defined by SHA-256, equal to 0x6a09e667bb67ae853c6ef372a54ff53a510e527f9b05688c1f83d9ab5be0cd19, and thus these 256 input bits are missing in the equations given.

I guess you'll have to rewrite (or get from another source) the equations of a SHA-256 round from the definition, and then check that when the 256-bit input from the previous round is set to the right constant, your work is equivalent to the equations given.