It was not clear that your example featured a 3/8 wavelength ground plane.
This makes it even more interesting.
Instead of the T-match tuner, a simple L-network at the base of the antenna performs a miracle.
The first configuration is to be preferred as it uses a smaller coil and hence has less loss.
E.g 16 turns 0.8 mm enammelled wire on a T-106-6 toroid will be fine.
The L-tuner presents 50 ohm to the coax feed-line.
The now matched 3.5 meter feed-line suffers a massive loss of 0.066 dB, instead of the previous 1.6 dB.
Assuming that the loss in the L-network tuner is similar to the loss of the T-match, we now have a grand total loss of 0.766 dB, which is a major improvement.
Discussing a QCX / QDX followed by the Elecraft T-1 automatic tuner, these examples show that the T-1 should be placed at the feed-point of the antenna.
For single band use, constructing a L-network tuner is quite easy. I use a similar arrangement on 40 m, matching an 8 meter long inverted-L, fed with 25 meter RG58 from my 40m QCX+ running 1W WSPR. Good enough for world wide coverage.
The bottom line: if you have to use an antenna tuner, its location can make a difference.
Roelof Bakker, pa0rdt