Software transactional memory (STM) has proven to be a useful abstraction for developing concurrent applications, where programmers denote transactions with an atomic construct that delimits a collection of reads and writes to shared mutable references. The runtime system then guarantees that all transactions are observed to execute atomically with respect to each other. Traditionally, when the runtime system detects that one transaction conflicts with another, it aborts one of the transactions and restarts its execution from the beginning. This can lead to problems with both execution time and throughput. In this paper, we present a novel approach that uses first-class continuations to restart a conflicting transaction at the point of a conflict, avoiding the re-execution of any work from the beginning of the transaction that has not been compromised. In practice, this allows transactions to complete more quickly, decreasing execution time and increasing throughput. We have implemented this idea in the context of the Manticore project, an ML-family language with support for parallelism and concurrency. Crucially, we rely on constant-time continuation capturing via a continuation-passing-style (CPS) transformation and heap-allocated continuations. When comparing our STM that performs partial aborts against one that performs full aborts, we achieve a decrease in execution time of up to 31% and an increase in throughput of up to 351%.