Hi Art n Barb
The one that jumps out is very close indeed to you in Deruta - Lungarotti in Torgiano, about 8 mins by car. They are a very good producer, albeit in a region where they've had to lead the way for quite a while, so no resting on the region's name. Always worth making an appointment and it may be essential (Tuscany is quite the exception in having many manned cellar doors / tasting room shops, the norm elsewhere is for visits by appointment). Don't let them be put off by this - email the winery, telling them which days would be convenient for your daughter and ask if it's possible to organise a visit.
Cortona might make a good mid-point, but despite lots of vineyards, many of these are contracted to chianti / Montepulciano producers, so not too many cellar doors. There are some though, plus a regional enoteca in the town.
Montepulciano is also worth considering. No appointments necessary to cruise the tasting rooms in this attractive hilltop town, with Crociani a recommendation, but plenty to taste at, but also visit the barrel rooms housing enormous historic 'botti' barrels containing many 1000s of litres of wine. If wanting to avoid the town, Daviddi in Stazione Montepulciano are a strong recommendation of a family operation producing very good wines unnoticed by the critics. More well known are Poliziano directly opposite, who do charge for the tasting, but take the option of tasting/tour/food and it is a good experience. The only downside is it's the only place in Italy where they attempted to pressure us to buy - I was utterly taken aback as nothing like it had ever occurred in Italy before. Hopefully a one-off. Nearby Avignonesi are brilliant and well worth arrranging an appointment, and if they offer a taste of their vin santo, then snatch their hand off. One of the most memorable wines I'd tasted, and horrifically I believe the price is fair (about €150-200 a half bottle). Their vino nobile is a lot cheaper, but very good. I'd assume appointments required for out of town places.
Hope this helps
Ian