It works! At least in principle . . .
Using RichD's suggestion that reed switches wired in parallel would trigger Trackmate to record segment times, I did a very crude "proof of concept" test. I stripped the insulation off a spare reed switch sensor cable and added a second reed to the circuit, clamping it to the exposed wire with alligator clips.

Passing a motor magnet over either reed switch would complete the circuit and trigger the timing program to record a lap. Moving the magnet across both switches in rapid succession triggered a quick time, as it would when a car travels through a segment of the track.

A typical lap time on my track is 7 seconds, but I set the minimum lap time to zero for this test. Here you can see that the program recorded a "lap time" of 2.01 seconds in the red lane (which has the modified sensor cable with two parallel reed switches) as I passed the magnet over the reed switches in succession.
So thanks to the great input and suggestions from the forum members, I now have what I believe to be an inexpensive and effective way to record interval times. The next step will be to install a series of reed switches in the lane that I want to use for segment testing, and then splice them into the cable. My track has six lanes, but the cable has provisions for eight lanes, so I will designate one of the unused lanes as my segment timer.
Incidentally, I also contacted Greg Braun, who created the Lap Timer 2000 freeware program and asked if there was a workaround that would allow me to do interval timing with his program using photo sensors. He replied that it wasn't possible.
Thanks!
Rick V.