We study the awake complexity of graph problems that belong to the class O-LOCAL, which includes a subset of problems solvable by sequential greedy algorithms, such as $(Δ+1)$-coloring and maximal independent set. It is known from previous work that, in $n$-node graphs of maximum degree $Δ$, any problem in the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity $O(\logΔ+\log^\star n)$. In this paper, we show that any problem belonging to the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity $O(\sqrt{\log n}\cdot\log^\star n)$. This leads to a polynomial improvement over the state of the art when $Δ\gg 2^{\sqrt{\log n}}$, e.g., $Δ=n^ε$ for some arbitrarily small $ε>0$. The key ingredient for achieving our results is the computation of a network decomposition, that uses a small-enough number of colors, in sub-logarithmic time in the Sleeping model, which can be of independent interest.