No, the context does not state any such alternative that if she wishes to keep her hair uncovered then she may do that by keeping them short or shaving completely. He is not suggesting an alternative between cutting hair or shaving bald; he says that for a woman to cut her hair short, or to shave bald, is very disgraceful. Having short hair or shaved hair are both disgraceful among Israelites of that period, although that is less disgraceful than displaying uncovered hair.
Frederick Louis Godet's commentary says
To impress the revolting character of such a course, the
apostle supposes it pushed to extremity. There is something of
indignation in his words: “If this woman has effrontery enough to do
the first of these acts, well and good, better also do the second!”
The repulsive character of the one should make that of the other felt.
The word ξυρασθαι is usually accented, as if it were the present
infinitive passive of ξυράω (ξυρᾶσθαι). But why should it not be
regarded as the aorist infinitive middle, like κείρασθαι, of the form
ξύρω (ξύρασθαι)? See Passow. There is a gradation from the one of
these verbs to the other: To cut the hair or even to shave the head.
The word αἰσχρόν, shameful, includes the two notions of physical
ugliness and moral indecency. (italics added)
Charles Ellicott's commentary says
(6) Let her also be shorn. —The force of this argument depends on the
fact that a woman’s head being uncovered would be regarded by others
as implying the same shame as was indicated by a woman’s hair being
cut short (i.e., shorn), or altogether removed (i.e., shaven). It is
as if the Apostle said —If a woman insists on her right to pray and
speak in an assembly with uncovered head, let her carry out this
principle to its logical result; let her insist on her right to have
her hair cut short, so as to show her equality with man —and what would
be thought of her then! No woman with a spark of shame in her would
think of doing that. Accordingly you admit that this principle of
sexual equality does not apply in all such matters; and it is
illogical to argue in favour of any general principle as if it were of
universal obligation, when you yourselves admit that it is not
applicable in some cases.