The question that comes first to my mind is, "Why do you care?"
Certainly an antenna does not need to be resonant to work well. I can
imagine you'd like a reasonably low indicated SWR, just so your
transmitter has a reasonable load to drive.

If you really want to know what's going on at the antenna feedpoint,
you'll need to back the effects of the feedline out of your antenna
analyzer readings, or use an analyzer that does it for you. If you
have a reasonable estimate of the feedline loss and know its electrical
length (easy to find if you put a short at the end of the line and look
at the resulting impedances read on the analyzer), then you should be
able to translate your analyzer readings to actual feedpoint impedance.
Do you have the feedline properly decoupled from the antenna so it's
not a significant part of the radiating system? If not, there seems
little reason to bother making the measurements.

I'd expect half-wave dipole resonance to result in lowest SWR on a
50-ohm feedline, but it won't be a very sharp minimum. So is it worth
worrying about?

Another 'speriment to try: build a fairly wide-spaced two wire
transmission line from your wire. Short it at one end, open at the
other, and look for quarter-wave resonance; or short both ends and look
for half-wave resonance. Measure the resonant frequency, which will be
a pretty sharp resonance (much sharper than the dipole). Remove the
insulation and see how much the resonance changes. Try for various
spacings to see what effect the spacing has. (Expect that close
spacings will show more effect than wide.)

Cheers,
Tom