In the pics: The late 19th/early 20 century practice of wearing a mantilla with the sari. Often made of lace. Probably a result of the popularity of the mantilla in nineteenth century Europe and more or less confined to Bengal.
Though we think of Gayatri Devi – and perhaps Indira Devi – as epitomising Indian elegance, the original style queen is without doubt Suniti Devi The mantilla was popularised by her (pic 1 via Getty), it was part of a simplification of the brahmika style introduced by Jnanadanandini Devi
Though not akin to lace mantillas, the practice of wearing an extra piece of cloth draped over the sari on the upper part of the body is common in weddings in some parts of India (e.g. the Marathi shela, also in this Telugu wedding and in Bengal) – pic 3 is fairly similar though I believe it is intended as an adjunct to the sari that falls right down to the lower leg.
pic 2: via Geraldine Forbes. pic 4: Alkazi collection
A modern version here.