Hi Clenaranjo,hello everyone,
Welcome in Tunisia,I would like to sweep your worry being a foreign women with your 3 years child regarding safety.You can go out your hotel by yourself.Regarding what you can do,depending on your interests and the location of your hotel there are many interesting places you may visit on your own some might be in walking distance for example if you'll be in Yasmine Hammamet,you can go to Medina Mediterranea ( medina.com.tn ) there are several attractions for your child such as Carthage land.
The old Medina in Hammamet center is worth a visit,Nabeul known for it's potteries,tiles and it's colorful Friday market is about 15 km for Hammamet center.Friga park (http://friguia-park.com/pages/en/accueil.php ) is also few kilometers from Hammamet .
You may join the organized sightseeing day tours to the must see near by Hammamet or do your sightseeing on your own.
You may allocate the 3 remaining days to visit Tunis, Carthage,Sidi Bou Said, go to Dougga ( a day trip ) if you're interested in archaeology or to Oudna,Thuburbo-Majus,Zaghouan
I wish that other foreign women who visited Tunisia on their own will give you their opinion to comfort you.
I hope that my input is helpful,feel free to ask me if you may need more details.I wish you a nice stay
There is a lovely garden between Yasmine Hammamet and Hammamet Centre. There is also lovely length of road behind Hammamet Medina named the kissing path ( no one does this). To be safe move around in daytime and do not display an over friendly face and ignore any peddlers. You can buy from places saying fixed price but can haggle if you but elsewhere. The souks and the safari like zoo are amazing but only go with yellow taxi.Also a pirate ship ride is great. Expect stray cats.
I have been to Hammamet several times alone but like everywhere else, be careful. Realistically, the culture is different and women who smile easily is a magnet for the unscrupulous. My advice is to cover yourself arms and legs whenever you're going to unfamiliar places to aviod each person coming with his wares.As Ben says Carthage Land is good but I don't know Tunis
Thank you for your words and for all the information, it is very useful and feel more comfortable now.