Genes in SARS-CoV-2 and other viruses in the order of Nidovirales are expressed by a process of discontinuous transcription mediated by the viral RNA-dependent RNA polymerase. This process is distinct from alternative splicing in eukaryotes and produces subgenomic RNAs that express different viral genes. Here, we introduce the DISCONTINUOUS TRANSCRIPT ASSEMBLY problem of finding transcripts T and their abundances c given an alignment R of paired end short reads under a maximum likelihood model that accounts for varying transcript lengths. Underpinning our approach is the concept of a segment graph, a directed acyclic graph that, distinct from the splice graph used to characterize alternative splicing, has a unique Hamiltonian path. We provide a compact characterization of solutions as subsets of non-overlapping edges in this graph, enabling the formulation of an efficient progressive heuristic that uses mixed integer linear program. We show using simulations that our method, JUMPER, drastically outperforms existing methods for classical transcript assembly. On short-read data of SARS-CoV-1, SARS-CoV-2 and MERS-CoV samples, we find that JUMPER not only identifies canonical transcripts that are part of the reference transcriptome, but also predicts expression of non-canonical transcripts that are well supported by direct evidence from long-read data, presence in multiple, independent samples or a conserved core sequence. Moreover, application of JUMPER on samples with and without treatment reveals viral drug response at the transcript level. As such, JUMPER enables detailed analyses of Nidovirales transcriptomes under varying conditions.