Go a step further with your Nano.
After getting the swr data, also get the impedance information (R +/- jX). You may find one case of high swr involves a fairly low impedance, while another case is associated with a fairly high impedance. Some tuners are happier matching one or the other. depending on their design. Especially if there is a large reactive (X) component of the impedance.
Report back what you find, and also report:
is there a feedline involved, or do the counterpoise and antenna both terminate at the rig?
how high is the far end of the antenna?
is the counterpoise a single conductor? how long?
how is the counterpoise deployed? on the ground? elevated? if so, how high?