  A candidate explanation for Early Mars rivers is atmospheric warming due to
surface release of H$_2$ or CH$_4$ gas. However, it remains unknown how much
gas could be released in a single event. We model the CH$_4$ release by one
mechanism for rapid release of CH$_4$ from clathrate. By modeling how
CH$_4$-clathrate release is affected by changes in Mars' obliquity and
atmospheric composition, we find that a large fraction of total outgassing from
CH$_4$ clathrate occurs following Mars' first prolonged atmospheric collapse.
This atmosphere-collapse-initiated CH$_4$-release mechanism has three stages.
(1) Rapid collapse of Early Mars' carbon dioxide atmosphere initiates a slower
shift of water ice from high ground to the poles. (2) Upon subsequent
CO$_2$-atmosphere re-inflation and CO$_2$-greenhouse warming, low-latitude
clathrate decomposes and releases methane gas. (3) Methane can then perturb
atmospheric chemistry and surface temperature, until photochemical processes
destroy the methane. Within our model, we find that under some circumstances a
Titan-like haze layer would be expected to form, consistent with transient
deposition of abundant complex abiotic organic matter on the Early Mars
surface. We also find that this CH$_4$-release mechanism can warm Early Mars,
but special circumstances are required in order to uncork 10$^{17}$ kg of
CH$_4$, the minimum needed for strong warming. Specifically, strong warming
only occurs when the fraction of the hydrate stability zone that is initially
occupied by clathrate exceeds 10%, and when Mars' first prolonged atmospheric
collapse occurs for atmospheric pressure > 1 bar.
