All genomes include gene families with very limited taxonomic distributions that potentially represent new genes and innovations in protein-coding sequence, raising questions on the origins of such genes. Some of these genes are hypothesised to have formed de novo, from non-coding sequences, and recent work has begun to elucidate the processes by which de novo gene formation can occur. A special case of de novo gene formation, overprinting, describes the origin of new genes from non-coding alternative reading frames of existing open reading frames (ORFs). We argue that additionally, out-of-frame gene fission/fusion events of alternative reading frames of ORFs and out-of-frame lateral gene transfers could contribute to the origin of new gene families. To demonstrate this, we developed an original pattern-search in sequence similarity networks, enhancing the use of these graphs, commonly used to detect in-frame remodelled genes. We applied this approach to gene families in 524 complete genomes of Escherichia coli. We identified 767 gene families whose evolutionary history likely included at least one out-of-frame remodelling event. These genes with out-of-frame components represent 2.5% of all genes in the E. coli pangenome, suggesting that alternative reading frames of existing ORFs can contribute to a significant proportion of de novo genes in bacteria.