There are necessary and sufficient conditions in the literature for a (left) partial order $\le$ on $G$ to extend to a (left) linear order $\le^{\ast}$ on $G$. This shows in particular, that not every partial left order extends to a linear left order in the non-abelian case, even though the group is orderable.
In the paper "Right-orderability of groups" by Richard Kaye (1998) these conditions are called "a sort of mini completeness/soundness theorem".
In the paper "Compactness of the space of left orders" (arXiv) of Dabkowska, Dabkowski, Harizanov, Przytycki and Veve, these conditions are referred to as Conrad's theorem (P. F. Conrad, Right-ordered groups, Mich. Math. J. 6(3), 1959, pp. 267–275.) This paper also gives an explicit example of a partial order on the fundamental group of the Klein bottle that does not extend, even though the group is orderable.
Remark: I have edited the answer to include Alexander's useful comments, which helped to clarify the answer.