Hi Philip,
I see the same thing. It can be eliminated by increasing the number of points plotted. But I do think it would be better if Plot knew to plot to the endpoints of segments.
Plot[{Piecewise[{{0.6 - 0.3692*(x + 3) - 0.0308*(x + 3)^3, -3 <
x <= -2}, {0.2 - 0.46154*(x + 2) - 0.0923*(x + 2)^2 +
0.95385*(x + 2)^3, -2 < x <= -1}, {0.6 + 2.2154*(x + 1) +
2.7692*(x + 1)^2 - 2.5846*(x + 1)^3, -1 < x <= 0}, {3 -
4.9846*x^2 + 2.58462*x^3,
0 < x <= 1}, {0.6 - 2.21538*(x - 1) + 2.76923*(x - 1)^2 -
0.95384*(x - 1)^3,
1 < x <= 2}, {0.2 + 0.46154*(x - 2) - 0.0923*(x - 2)^2 +
0.0308*(x - 2)^3, 2 < x <= 3}}],
3 - 3.20667*x^2 + 0.866667*x^4 - 0.06*x^6}, {x, -3, 3},
PlotPoints -> 1000]