The synergistic production of gas hydrates and deep gas reservoirs assisted by the geothermal energy transport via circulating water is significantly influenced by the temporal coordination of reservoir production schedules. However, the correlation between production timing and system performance remains unexplored. This work has for the first time systematically demonstrated the decisive role of choosing the production timing in determining the production efficiency. Four combined extraction methods for the co-operation of hydrate reservoirs and deep gas reservoirs have been designed. Experimental results demonstrate that deep gas reservoirs maintain high productivity independent of hydrate decomposition dynamics. This finding highlights the critical role of deep gas production timing in modulating geothermal energy transport to accelerate hydrate dissociation. Strategic delay of deep gas production until after initial hydrate reservoir depressurization was found to elevate circulating water temperatures, thereby expanding the heat distribution within the hydrate reservoir. This approach increased hydrate production efficiency by 6.79% compared to simultaneous production of both reservoirs. To maximize this thermal enhancement effect, deep gas production was further delayed to the early and middle stages of the hydrate reservoir's constant-pressure phase. Results revealed that sustained high-temperature water circulation until the mid-constant-pressure stage could boost cumulative geothermal energy release, achieving a 10.49% further improvement in hydrate production efficiency. These findings establish a positive correlation between deep gas production timing and hydrate recovery performance. The study provides actionable insights for optimizing field-scale co-production strategies, where deliberate sequencing of reservoir production can significantly enhance geothermal-driven hydrate dissociation rates.