General view of the OPERA Detector
Image from the OPERA experiment.
Picture of the MINOS detector in the Soudan Underground Mine State Park, during (re)construction.
Image courtesy of Jon 'ShakataGaNai' Davis
Answer: Using another neutrino detector is not feasible right now. Neutrino detectors are not scattered plentifully about the landscape. They are complicated pieces of equipment that have to be shielded in underground laboratories. The OPERA experiment that made this measurement only has one of them at its disposal.
However, a version of your idea is actually being used by another neutrino experiment, MINOS, based in Fermilab outside of Chicago. They have two neutrino detectors, one at Fermilab itself (the “near” detector) and one in a mine about 450 miles away (the “far” detector). They are therefore able to measure the time difference between detection of the beam in the near and far detectors, so in principle they can determine both the departure and arrival times of the neutrinos. The MINOS experiment will probably be the one to replicate the OPERA experiment and see if they, too, measure a speed faster than light.