Activated carbons are widely used for sustainable technology of adsorptive transformation and storage of heat. Here, we analyze the applicability of twelve commercial carbons and an innovative carbonaceous composite “LiCl confined to multi-wall carbon nanotubes” (LiCl/MWCNT) for a new cycle “Heat from Cold” (HeCol). It has recently been proposed for amplification of low- temperature ambient heat in cold countries. The analysis is made in terms of the methanol mass exchanged and the useful heat generated per cycle; the latter is the main performance indicator of HeCol cycles. The maximum specific useful heat, reaching 990 and 1750 J/g, can be obtained by using carbon Maxsorb III and the composite, respectively. For these materials, methanol adsorption dynamics under typical HeCol conditions are experimentally studied by the large pressure jump method. Before making this analysis, the fine carbon powder is consolidated by either using a binder or just pressing to obtain larger particles (ca. 2 mm). The methanol desorption from the consolidated samples of Maxsorb III at T = 2 °C is faster than for LiCl/MWCNT, and the maximum (initial) useful power reaches (2.5–4.0) kW/kg sorbent. It is very promising for designing compact HeCol units utilizing the carbon Maxsorb III.