My wife is, Alhamdulillah a woman of purdah.

Answer

How can a purdahnasheen lady ever want to be in the car with a ghair mahram present? From the request of your wife, you will be able to understand the flimsiness of the hijaab/purdah of so-called hijaabi and purdahnasheen women of today. For them the cloak and perhaps a niqaab with it, are the be all of purdah. There is no genuine purdah in the heart which is the headquarters of Purdah. The purdah of the vast majority of women of today is a mere outer façade to advertise themselves as women of the Deen. Their purdah is either traditional or tablighi style, but not Shar'i.

Traditional 'purdah' is merely a mark of respect or to convey that the women is from a 'good', 'respectable' family of high or good lineage. This is the type of purdah which the vast majority of so-called 'pious', 'Deeni conscious' families observe. Their 'purdah' is for public consumption and has its base in riya. This is the type of traditional purdah which almost 100% of the so-called 'muttaqi' ulama and even hazrat's and 'shaikhs' of today observe. Thus, within the confines of their homes men, molvis, and buzroogs freely mix with sisters-in-law, female cousins, wives of uncles and the like.

'Tablighi' purdah is another huge fraud which is tantamount to kufr. In the name of the Deen these wayward miscreants commit the kufr of halaalizing female safar (travel) without mahrams. For these tablighi miscreants, be they hazrats and buzrugs, the determinant of their deeds is not Allah's Shariah. It is their lopsided understanding of Tabligh. What appears to be 'tabligh' to their nafs, is adequate daleel for scuttling the ahkaam of the Shariah. The outcome of all their haraam ghulu has been the spawning of two mutually hostile factions set to develop into deviant sects. The Ulama need to watch the direction of these sects. Tablighi 'purdah' is also confined to the cloak and niqaab while the heart is practically bereft of hijaab.