Adjoining greatest and smallest elements to uncountable ordered field

Question

As titled. With $\mathbb{R}$, we can adjoin $+\infty$ and $-\infty$ as the greatest and smallest elements and define $\overline{\mathbb{R}}:=[-\infty, +\infty]$. But if we have an ordered field $\mathbb{F}$ with a cardinal number larger or equal to $2^{\aleph_1}$, can we still do similar things and define $\overline{\mathbb{F}}$? I only know of well-orderings and the definition of ordinal numbers. But $\mathbb{F}$ is not well-ordered. Hence the question. Thanks in advance.