Complex sentence: The new pairing got work playing as an in-flight band on Virgin Atlantic flights to and from the United States--after sending some demo material to Nashville-based label MCA, the band was called six weeks later to record their first album.
Split into simpler sentences: The new pairing got work as an in-flight band on Virgin Atlantic flights to and from the United States. Six weeks after sending demo material to Nashville-based label MCA, the band was called to record their first album.