A device (1) having an auxiliary handrail (2) under an existing main handrail (3) and bridge members (4) each including: a spacer member (5), an upper coupling member (6) and a lower coupling member (7). The upper coupling member includes a tension strap (8), which by means of two connection points (9, 10) is fastened to the upper coupling member, and which is arranged for clamping the upper coupling member against the main handrail. Furthermore a displacement/force-transformation-based strap-tensioning device is provided, for example a screw-tensioning member, whereby the upper coupling member is forcibly and accurately adjustably clamped against the main handrail.