no, the word you use as cover, what is it in arabic and what is the meanin in arabic?
also there is the verse O Prophet! Enjoin your wives, your daughters, and the wives of true believers that they should cast their outer garments over their persons (when abroad): That is most convenient, that they may be distinguished and not be harassed
in arabic that is Ya ayyuha alnnabiyyuqul li-azwajika wabanatika wanisa-ialmu/mineena yudneena AAalayhinna min jalabeebihinna thalikaadna an yuAArafna fala yu/thayna
you can clearly see the word jalabeeb, which is the plural of jilbaab.
and you have to take it in the cultural context as well,
if i ask a sweedish person to put their scarf on they will put somethin around their neck,
if i ask a british pakistani they would put somethin on their head, so you cant take a word out of context of its cultural meanin and interpret it to your own desires althouh all scholars and reliious roups, knowledable people etc are in areement on this issues.
their are many areas of disareement in islam which are valid and all opinions equally valid, but this is not one.