Can I get more information from this signal other than magnitude?
There's a lot of answers -- look at what's different between the metal and no-metal case, and try to make some signal processing that'll do the best job of differentiating between them.
Something that I suspect that will work well would be to start a clock as soon as the pulse goes negative (possibly according to the transmitter timing, if that will be reliable).  Capture the height of the peak, then integrate the area of the whole response over time.  That peak to area ratio should be very indicative of the presence of metal.
My gut feel is that the ratio of the peak to the integral of the overall response should be more indicative than the actual value of the integral -- but if the pulses are repetitive enough, just that integral may be enough.  If you want to go down that road, be sure to try to detect every different thing you may be concerned with, at every different range.
It may be more reliable to capture the peak by integrating for a short time, and comparing that number to the integral of a longer time span.
Continuing in this direction, if you were to integrate the pulse over time and look at the shape of the resulting integrated pulse it should be very indicative of what's going on.  Once again my gut feel is that you want to look at ratios -- I'd normalize the integrated pulse to its final value.  The points at which it reaches 1/4, 1/2, 3/4, etc., of its final value should be very telling.