For 100% of hyperelliptic curves z^2 = f(x,y) ordered by height, every odd-degree algebraic point of degree at most 2g-1 has Weil height at least (1 + 1/(2g+2) - epsilon) log Ht(f).
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Lower bounds on heights of odd degree points of hyperelliptic curves
For 100% of hyperelliptic curves z^2 = f(x,y) ordered by height, every odd-degree algebraic point of degree at most 2g-1 has Weil height at least (1 + 1/(2g+2) - epsilon) log Ht(f).