Really? Since when did the solstice lasted several weeks or months.
The soltice (from the Latin for “sun standing still”, in its endless north-south oscillation) is but an instant. In fact, near the solstice, the motion of the sun is slow enough that its motion in a day is probably much less than the resolution of the instruments available.
Presumably the Greeks knew the dates of the soltices well enough, but you would also have to know the direction south (the direction of the meridan) and have a way of measuring the sun’s altitude.
But anyone who knew enough positional astronomy to know the date of the solstices and had sufficient instrumentation to measure the sun’s altitude when it crossed the meridan, could have determined altitude from a few key stars much more easily. (Polaris was not the Pole Star in those days, nor was there a good alternative.)
The sun is possibly the worst choice of a reference for latitude. It must have occured to people who navigated on open waters that the stars provide a good reference frame.