It sounds as though you need underlining to add structure rather than interlining to add warmth.
From "Threads Sewing Guide":
Underlining and interlining are both interior layers of fabric placed beneath the fashion fabric. The terms are sometimes used interchangeably, but the functions of interlining and underlining are different.
Underlining is placed behind the fashion fabric to improve the look, feel and functionality of the the fabric. It provides body, opacity and support for longer wear and a more professional appearance. Construction details can be attached to the underlining without evidence of them showing on the garment outside. Typical underlining fabrics include cotton batiste, cotton flanelette, wool flannel or felt, rayon challis or, for nearly weightless support, silk organza.
Interlining is a layer of fabric added to the garment lining for warmth and insulation. Cotton flannel is a favourite for interlining, but you can use wool or synthetic batting that's made specifically for this purpose; these add more warmth with less weight.