Despite large-scale investigations of homogeneous single-site metallocene catalysts and systems based on them, there are still unsolved problems related to the control of their activity and chemo- and stereoselectivity. A solution to these problems is required to develop efficient methods for the synthesis of practically useful products of alkene transformations, such as dimers, oligomers, and polymers. Here we studied the catalytic activity of structurally diverse zirconocenes (L2ZrCl2, L = Cp, C5Me5, Ind, L2 = Me2CCp2, Me2SiCp2, Me2C2Cp2, rac-Me2CInd2, rac-H4C2Ind2, BIPh(Ind)2, H4C2[THInd]2), and co-catalysts activating the system, namely HAlBui2, MMAO-12, and (Ph3C)[B(C6F5)4], at low activator/Zr ratios in a 1-hexene oligomerization reaction. The influence of catalyst structure and system composition on the alkene conversion, the type of products, and the reaction stereoselectivity were investigated. The composition of hydride intermediates formed in the L2ZrCl2–HAlBui2–activator system (L2 = ansa-Me2CCp2, Ind) was studied by NMR spectroscopy. Participation of the bis-zirconium hydride complex as the precursor of catalytically active sites of the alkene dimerization reaction was shown.