Unless you can complete your run between first light and sunrise, I would tend to the view that it is probably better to run in late afternoon, so that you can then rehydrate pretty much as soon as you are finished. Risking going the whole day dehydrated is probably not wise, and if you are not feeling you have had enough to drink, you can always skip going running that day.
Also, splashing some water over yourself before/during your run can help keep you cool, and sweating less, and thus reduce the amount of water you need to drink. (I don't know the specifics of the religious rules around Ramadan, but I assume this would not break them).
Another option would be to plan on walking most of the time, but if there is a cooler day, sieze the opportunity to go running then. Even running once a week or so is probably enough to at least 'keep your hand in'.
The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.
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