Subwoofer hook up

I have a pretty basic question but I keep getting different answers from people. I have an AVR for movies and an integrated amp with a home theater bypass for when I listen to music. I am looking to replace my REL Storm III with a new subwoofer. The Storm uses a "Neutrik" connector which may be proprietary to REL. It connects to the subwoofer directly from the speaker outputs on the amp. A new subwoofer will not use this connector and I have the option of connecting it via the AVR or via the pre-outs of the amp. Which is correct. I want to listen to music without turning on the AVR (but obviously using the sub) and I need the AVR when watching movies. I really don't want multiple cables and switching devices as some have also suggested.

Thanks