Non-mechanical switching device?

[Hermansonsc]

You might want to design it as a make-before-break circuit so there is no time at which nothing is connected.  This connects both inputs together briefly so there is no audible switching noise when transitioning from one state to the other.  This is done by adding R-C networks that delay turnoff but not turn on.

I understand the concept, but I have no idea how to achieve it. What would that require in this type of circuit?

[mimmotronics]

The solution PRR and Mark Hammer suggested sounds to me like the most logical solution to your particular case. To build it yourself, you need only 4 jacks, two for input/output and two for send/return. That can easily fit in a 1590B. Wire up the switch the same way you would if you were implementing a true bypass on an existing stompbox.

For a more complex example I posted a pic on my instagram of a 5-loop looper you can check out. (I'm not sure how to post pics here...) It's a bit more complex but the idea is the same, just multiplied by 5