We studied solvent-driven ordering dynamics of block copolymer films supported by a densely cross-linked polymer network designed as organic hard mask (HM) for lithographic fabrications. The ordering of microphase-separated domains at low degrees of swelling corresponding to intermediate/strong segregation regimes was found to proceed significantly faster in films on a HM layer as compared to similar block copolymer films on silicon wafers. The 10-fold enhancement of the chain mobility was evident in the dynamics of morphological phase transitions and of related process of terrace formation on a macroscale as well as in the degree of long-range lateral order of nanostructures. The effect is independent of the chemical structure and on the volume composition (cylinder/lamella forming) of the block copolymers. In-situ ellipsometric measurements of the swelling behavior revealed a cumulative increase in 1-3 vol % in solvent uptake by HM-block copolymer bilayer films, so that we suggest other than dilution effect reasons for the observed significant enhancement of the chain mobility in concentrated block copolymer solutions. Another beneficial effect of the HM-support is the suppression of the film dewetting which holds true even for low molecular weight homopolymer polystyrene films at high degrees of swelling. Apart from immediate technological impact in block copolymer-assisted nanolithography, our findings convey novel insight into effects of molecular architecture on polymer-solvent interactions.