Your question contains a contradiction because you propose:
elementary point particles, in contact with each other
but by definition point particles have zero size and cannot be placed in contact with each other. As far as we know elementary particles are indeed pointlike, so you can't have a rod of a finite length made up from elementary particles in contact.
Your rod would have to be made up from extended particles like atoms or neutrons, and these have a non-zero size because they are a bound state of elementary particles. However the size of an extended particle is Lorentz contracted in exactly the same way as the rod is, so your rod would shrink at the same rate as the (extended) particles that make it up shrink.
For completeness I should point out that the Lorentz contraction is not really a contraction but a rotation. More precisely it's a hyperbolic rotation in 4D spacetime. In the moving coordinate system the ends of the rod rotate so they are at different times. The rod looks contracted because the observers in the rod rest frame and moving frame disagree about the times they are observing the ends of the rod. For a detailed discussion of this see "Reality" of length contraction in SR.