What you wear outside the house is irrelevant as long as it is according to the rules of shari'a (which means she should try to hide all of her body except with hands and face - this is the view on which there is consensus, some scholars may allow he to show her feet while others may ask her to hide her face too, but those are matters of fiqh or scholarly disputes).
Also you should know that woman might even need to wear hijab inside the house if there are foreign visitors. As verse (24:31) is a general order and not specific for covering outside the house.
Note that the so called hijab verses seem explicitly indicating that this apply for the case of them going outside the house:
O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments. That is more suitable that they will be known and not be abused. And ever is Allah Forgiving and Merciful. (33:59)
For details read my answer on Did women in Arabian peninsula have "Hijab" before Islam?
This verse seems to be have a direct relation to (33:33) showing that it is more preferable to stay at home. So covering outside the house is therefore more emphasized by (33:59).