Small point of confusion, if $S[\Lambda]$ is the Dirac spinor rep, and I want to find the adjoint of this operator $(S[\Lambda]^\dagger)$, I take the adjoint of the exponential:
$$\exp{-\frac{i}{2}\omega_{\mu\nu}(J^{\mu\nu})^\dagger}$$
and use the identity $(S^{\mu\nu})^\dagger=\gamma^0 S^{\mu\nu}\gamma^0$, that's fine, but we then exponentiate the product $\gamma^0 S^{\mu\nu}\gamma^0$, which Tong says becomes $\gamma^0 S[\Lambda]\gamma^0$, and the only way I can imagine this working is if we Taylor expand the exp and pull out factors of $\gamma^0$ on either side (using $\gamma^0=\Bbb 1$ …