Overlay broadcast is fundamental to large-scale distributed systems, enabling messages to be disseminated to all participants. Plumtree is a well-established protocol that embeds a spanning tree in a gossip overlay: payloads are disseminated along tree links, while non-tree links support recovery and optimization. However, because the tree is constructed from the first broadcast, it is optimized for the initial source and can introduce long detours for broadcasts originating at other nodes, particularly leaf nodes. To address this limitation, we propose probabilistic exuding, an off-tree forwarding mechanism in which nodes near the source additionally forward payloads over their non-tree links with a specified probability. Exuded messages use a dedicated message type and do not participate in tree construction or repair, leaving the shared tree unchanged. Simulations on a 1000-node random regular overlay show that, for randomly selected sources and a representative parameter setting, the proposed method reduces the 95th-percentile delivery latency by approximately 40% relative to Plumtree. This closes approximately 92% of the latency gap between Plumtree and flooding while sending only about 58% as many payload messages as flooding. In contrast, Plumtree's own link optimization and shorter repair timeouts, which also exploit non-tree links, rewire the shared tree and increase latency as broadcast sources change; the latency of exuding is nearly unaffected.

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