  In this paper we develop a technique for proving determinacy of classes of
the form $\omega^2-\Pi^1_1+\Gamma$ (a refinement of the difference hierarchy on
the co-analytic sets lying between $\omega^2-\Pi^1_1$ and
$(\omega^2+1)-\Pi^1_1$) from weak principles, establishing upper bounds for the
determinacy-strength of the classes $\omega^2-\Pi^1_1+\Sigma^0_\alpha$ for all
computable $\alpha$ and of $\omega^2-\Pi^1_1+\Delta^1_1$. This bridges the gap
between previously known hypotheses implying determinacy in this region.
