No, because all the off-rectilinear parts are destructively interfered away.
Huygen's principle is often stated differently than how Huygens stated it. He said that if you draw a sphere at every point of the wavefront, the future wavefront is the envelope of the spheres so drawn, and only one of the envelopes (the one that keeps going forward). So from your sphere, you are supposed to imagine drawing a bunch of other spheres, and then looking at the outermost limit of where they reach. This is a larger sphere, of radius ct, and this is the new wavefront.
The modern version of Huygen's principle (I learned from this site) is the fact that the Green's function of the wave equation in 3+1 dimension is a delta-function on the light cone. This is a way of saying that the full propagation of the wave is by superposing the circles from propagating each point on the previous front. The superposition though is cancelling away from the region where the nearby circles have a mutual tangent, and this is the envelope criterion.