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Our method also improves the best lower bounds for $r(C_{\\ell},t)$ obtained by Bohman and Keevash from the random $C_{\\ell}$-free process by polylogarithmic factors for all odd $\\ell \\geq 5$ and $\\ell \\in \\{6,10\\}$. For $\\ell = 4$ it matches their lower bound from the $C_4$-free process.\n  We also prove, via a different approach, that $r(C_5, t)> (1+o(1))t^{11/8}$ and $r(C_7, t)> (1+o(1))t^{11/9}$. 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