Local icosahedral order is found to increase as random hard-sphere packings (one- and two-component) generated on the computer are densified from the (recently established) ``loose random-packing'' limit to the ``dense random-packing'' limit. While icosahedral ordering in ``atomic'' systems is frequently ascribed to the energetic stability of icosahedral clusters, the present results show that icosahedral ordering can arise from packing constraints alone. However, the icosahedra are often distorted, partly due to the lack of preferred distance between hard spheres. At high density one-third to one-half of the pairs in the first peak of the radial distribution function (RDF) are icosahedral fragments. The splitting of the second peak, which is characteristic of packings of spherical particles, was studied by decomposing the RDF into components according to the local environment of the pairs. Linear trimers of spheres are responsible for the second subpeak while the first subpeak arises roughly equally from tetrahedra sharing a face and triangles with adjacent sides. The hard-sphere packings were compared with packings of soft, attracting spheres by relaxing the configurations under a Lennard-Jones potential. The fraction of pairs characteristic of local crystalline order in the first peak of the RDF was found to increase. The reversal of the relative height of the two parts of the split second peak results from a broadening of the distribution of distances within the linear trimers, while the distribution sharpens for the face sharing tetrahedra and adjacent triangles.