I think there were already a couple of responses, so let me add a bit as well.
What's your integrated amp, and does it have a Pre-Out / Main-in set of sockets on the back? Frequently if it exists it will have a connector between the two sockets like this:
This is the link between the preamp of the integrated and the power amp side of the integrated.
If they exist, run the pre-out into the inputs on V3 and then the output of the V3 to the main-in/power-in.
In this case you will have two volume controls. I would set the V3 to max, and just use the integrated's volume as you do now. (I don't think that's an issue for the tubes, but
@bcowen or ACP could maybe suggest if that's an issue). If it is a concern for the tubes, reverse the idea, or even some combination of the two controls. I doubt you'd hear any signal degradation either way.
If you don't have the pre-out/main-in option, then without another external switch box, you could only run the V3 on one of your sources. So perhaps the DAC feeding the integrated. Run the output of the DAC into the V3 and V3 output into the input you currently use for the DAC. You will again have the two volume controls to contend with, but now the V3 is only playing with (in this example) the DAC as the source. This one does have the advantage that if your main source comes via the DAC, then you get DAC --> V3 --> Headphones, with no need to turn the integrated on.
You might need to match the gain on the V3 to the integrated, so always start with the low gain and adjust as needed.
And, in either case, no output to speakers with headphones as the V3 mutes the pre-out when headphones are connected.
