The famous Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. Kratochvil et al. applied this technique to prove that a k-CNF in which each variable appears at most 2^k/(ek) times is always satisfiable [KST93]. In a breakthrough paper, Beck found that if we lower the occurrences to O(2^(k/48)/k), then a deterministic polynomial-time algorithm can find a satisfying assignment to such an instance [Bec91]. Alon randomized the algorithm and required O(2^(k/8)/k) occurrences [Alo91]. In [Mos06], we exhibited a refinement of his method which copes with O(2^(k/6)/k) of them. The hitherto best known randomized algorithm is due to Srinivasan and is capable of solving O(2^(k/4)/k) occurrence instances [Sri08]. Answering two questions asked by Srinivasan, we shall now present an approach that tolerates O(2^(k/2)/k) occurrences per variable and which can most easily be derandomized. The new algorithm bases on an alternative type of witness tree structure and drops a number of limiting aspects common to all previous methods.