  We present baseband radio observations of the millisecond pulsar
J1909$-$3744, the most precisely timed pulsar, using the MeerKAT telescope as
part of the MeerTime pulsar timing array campaign. During a particularly bright
scintillation event the pulsar showed strong evidence of pulse mode changing,
among the first millisecond pulsars and the shortest duty cycle millisecond
pulsar to do so. Two modes appear to be present, with the weak (lower
signal-to-noise ratio) mode arriving $9.26$ $\pm 3.94$ $\mu$s earlier than the
strong counterpart. Further, we present a new value of the jitter noise for
this pulsar of $8.20 \pm 0.14$ ns in one hour, finding it to be consistent with
previous measurements taken with the MeerKAT ($9 \pm 3$ ns) and Parkes ($8.6
\pm 0.8$ ns) telescopes, but inconsistent with the previously most precise
measurement taken with the Green Bank telescope ($14 \pm 0.5$ ns). Timing
analysis on the individual modes is carried out for this pulsar, and we find an
approximate $10\%$ improvement in the timing precision is achievable through
timing the strong mode only as opposed to the full sample of pulses. By forming
a model of the average pulse from templates of the two modes, we time them
simultaneously and demonstrate that this timing improvement can also be
achieved in regular timing observations. We discuss the impact an improvement
of this degree on this pulsar would have on searches for the stochastic
gravitational wave background, as well as the impact of a similar improvement
on all MeerTime PTA pulsars.
