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.

Gupta, Narain; Rhemtulla, Akbar On ordered groups. Algebra Univ. 1 (1971/72), 129-132.Possibly more information can be found in Rhemtulla's book, but I haven't yet had access to it. $\endgroup$