A lot of research has focused on the efficiency of search engine query processing, and in particular on disjunctive top-k queries that return the highest scoring k results that contain at least one of the query terms. Disjunctive top-k queries over simple ranking functions are commonly used to retrieve an initial set of candidate results that are then reranked by more complex, often machine-learned rankers. Many optimized top-k algorithms have been proposed, including MaxScore, WAND, BMW, and JASS. While the fastest methods achieve impressive results on top-10 and top-100 queries, they tend to become much slower for the larger k commonly used for candidate generation. In this paper, we focus on disjunctive top-k queries for larger k. We propose new algorithms that achieve much faster query processing for values of k up to thousands or tens of thousands. Our algorithms build on top of the live-block filtering approach of Dimopoulos et al, and exploit the SIMD capabilities of modern CPUs. We also perform a detailed experimental comparison of our methods with the fastest known approaches, and release a full model implementation of our methods and of the underlying live-block mechanism, which will allows others to design and experiment with additional methods under the live-block approach.