Mark, congratulations! As Tony says, you are in for a treat.
I'm pretty certain that the output connectors on the SRM-006t are internally wired directly to the corresponding input connectors, for "input 1," and are provided simply as a convenience in case you want to route the signals provided to "input 1" to an additional destination. So there is no need to connect those output connectors to anything.
Assuming that you will want to be able to use the headphones to listen to the tape decks as well as to other source components that are connected to the preamp, what I suggest is that you connect an input of the SRM-006t to whichever tape output of the Niles (1 or 2 or 3) is presently unused.
To listen with the headphones, you would then set the "Play" switch on the Niles to "Off," and if (for example) the input of the SRM-006t is connected to tape out 3 on the Niles, you would set the Tape 3 knob on the Niles to either "Amp" or "1" or "2" depending on what you want to listen to. The "Amp" setting would be used to listen to whatever source component is selected on the preamp. "1" and "2" would be used to listen to the corresponding tape deck. The Tape 1 and Tape 2 knobs would be set to "Off" in all of these situations, assuming you are not recording while listening.
At some point you might want to compare the resulting sound with the sound that would result if you connect the SRM-006t directly to your most critical source component, as Tony suggested. I wouldn't be surprised, though, if the difference is very minimal.
Enjoy! Regards,
-- Al