The object of wearing a head-covering is, as quoted in your source, so that we may be imbued with the fear of G-d.
I suggest that the need to be imbued with the fear of G-d is independent of the weather conditions.
Further it should be possible to choose a head-covering which will deal adequately with the weather conditions.
But bear in mind that Shamash.org amongst other sources says:
Many Ashkenazi rabbis acknowledge that wearing a head covering at all
times was once considered an optional "midat chasidut" [pious act] but
that nowadays, full-time head covering is the norm except under
Sephardic communities generally did not have the custom of wearing a
kipa all the time.
Some diaspora Jews leave off the kipa at school, work, or when
testifying in court, because of real danger or uneasiness in appearing
in the secular world with an obvious symbol of Jewishness.
So, ideally one should have the head covered all the time. Extenuating circumstances can provide an exception but the protection for the fear of G-d will not be there under those circumstances.