LSM-based key-value (KV) stores are an important component in modern data infrastructures. However, they suffer from high tail latency, in the order of several seconds, making them less attractive for user-facing applications. In this paper, we introduce the notion of compaction chains and we analyse how they affect tail latency. Then, we show that modern designs reduce tail latency, by trading I/O amplification or require large amounts of memory. Based on our analysis, we present vLSM, a new KV store design that improves tail latency significantly without compromising on memory or I/O amplification. vLSM reduces (a) compaction chain width by using small SSTs and eliminating the tiering compaction required in L0 by modern systems and (b) compaction chain length by using a larger than typical growth factor between L1 and L2 and introducing overlap-aware SSTs in L1. We implement vLSM in RocksDB and evaluate it using db_bench and YCSB. Our evaluation highlights the underlying trade-off among memory requirements, I/O amplification, and tail latency, as well as the advantage of vLSM over current approaches. vLSM improves P99 tail latency by up to 4.8x for writes and by up to 12.5x for reads, reduces cumulative write stalls by up to 60% while also slightly improves I/O amplification at the same memory budget.