Original sentence: Climates during the Tertiary slowly cooled, starting off in the Paleocene with tropical-to-moderate worldwide temperatures and ending up with extensive glaciations at the end of the period.The Tertiary covered the time span between the superseded Secondary period and an out-of-date definition of the  Quaternary period.The Tertiary is a term for a geologic period 65 million to 1.8 million years ago.The earliest recognizable hominoid relatives of humans, Proconsul and Australopithecus, also.The period began with the demise of the non-avian dinosaurs in the Cretaceous–Tertiary extinction event, at start of the Cenozoic era,  spanning to beginning of the most recent Ice Age, at the end of the Pliocene epoch
Edited sentence: Climates during the Tertiary slowly cooled, starting off in the Paleocene with tropical-to-moderate worldwide temperatures and ending before the first extensive glaciation at the start of the Quaternary.The Tertiary covered the time span between the superseded Secondary period and the Quaternary.The Tertiary is a term for a geologic period 65 million to 2.588 million years ago.The earliest recognizable hominoid relatives of humans such as Proconsul.The period began with the demise of the non-avian dinosaurs in the Cretaceous–Tertiary extinction event, at start of the Cenozoic era, spanning to beginning of the most recent Ice Age, at the end of the Pliocene epoch