The low thermal conductance of polymers is one of the major drawbacks for many polymer-based products. However, a single polymer chain when stretched can have high thermal conductivities. We use non-equilibrium molecular dynamics simulations to study the steady-state thermal conductance along finite macromolecules under mechanical control of the end-to-end distance. We find that the nature of heat transport along such chains strongly depends on mechanical tuning, leading to significantly different heat conductions and temperature profiles along the chain in the compressed-chain and stretched-chain limits. This transition between modes of behaviors appears to be a threshold phenomenon: at relatively small end-to-end distances, the thermal conductance remains almost constant as one stretches the polymer chain. At given critical end-to-end distances, thermal conductances start to increase, reaching the fully extended chain values. Correlated with this behavior are two observations: first, the temperature bias falls mostly at contacts in the fully stretched chain, while part of it falls along the molecule in the compressed limit. Second, the heat conduction does not change significantly with the chain length in the stretched-chain limit but decreases dramatically when this length increases in the compressed molecule. This suggests that heat transfer along stretched chains is mostly ballistic, while in the compressed chain, heat is transferred by diffusive mechanisms. Significantly, these trends persist also for a large range of molecular structures and force fields, and the changing behavior correlates well with mode localization properties. Similar studies conducted with disordered chains and bundles of several chains show remnants of the same behavior.