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Joel David Hamkins
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Let me address merely the suggestion the OP makes in the comment: whether this ordinal can be specified from the cofinality of the field.

The answer is no, because any ordered field $F$ can be elementarily extended to a field with cofinality $\omega$, or indeed, to a field with any given regular cofinality. To have cofinality $\delta$, simply extend $\delta$ many times, making sure to put a new element on top each time, taking unions at limit stages.

$$F\prec F_1\prec F_2\prec\dots\prec F_\alpha\prec\dots\prec F_\delta$$

In particular, there can be fields with very large embedded ordinals, as large as desired, which still have cofinality $\omega$.

Joel David Hamkins
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