The 31-year-old went into the 800m round of the heptathlon needing to beat Belgian superstar Nafi Thiam by 8.5 seconds for gold.
And while she managed a personal best of 2:04.90, Thiam crossed the line within six seconds to claim her third successive gold.
It was still a first Olympic medal for KJT in a fourth Games.
The Liverpudlian won her second World Championship last year, having returned from a career-threatening Achilles injury in 2020, then pulling up with a calf tear in the Tokyo Olympics.
She led the heptathlon overnight but Thiam overtook her with a powerful javelin throw, eventually taking gold by 6880 points to 6844.
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