Older texts would use the appropriate grammatical gender for possessive pronouns, eg,
'Das Weib hat seinen Hut verloren.'
Is this still the best usage, or should we now use the natural gender, ie,
'Das Weib hat ihren Hut verloren'
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You'd still say
Das Weib hat seinen Hut verloren
although it's very uncommon today to refer to a women with "Das Weib". You would use "Die Frau" instead.
Das Mädchen hat seinen Hut verloren
would be grammatically correct, although you'll frequently hear
Das Mädchen hat ihren Hut verloren
in colloquial language.