In recent years, the interest in the Internet of Things (IoT) has been growing because this technology bridges the gap between the physical and virtual world, by connecting different objects and people through communication networks, in order to improve the quality of life. New IoT wearable devices require new types of antennas with unique shapes, made on unconventional substrates, which can be unobtrusively integrated into clothes and accessories. In this paper, we propose a fully textile dual-band logo antenna integrated with a reflector for application in IoT wearable devices. The proposed antenna’s radiating elements have been shaped to mimic the logo of South-West University “Neofit Rilski” for an unobtrusive integration in accessories. A reflector has been mounted on the opposite side of the textile substrate to reduce the radiation from the wearable antenna and improve its robustness against the loading effect from nearby objects. Two antenna prototypes were fabricated and tested in free space as well as on three different objects (human body, notebook, and laptop). Moreover, in the two frequency ranges of interest a radiation efficiency of 25–38% and 62–90% was achieved. Moreover, due to the reflector, the maximum local specific-absorption rate, which averaged over 10 g mass in the human-body phantom, was found to be equal to 0.5182 W/kg at 2.4 GHz and 0.16379 W/kg at 5.47 GHz. Additionally, the results from the performed measurement-campaign collecting received the signal-strength indicator and packet loss for an off-body scenario in real-world use, demonstrating that the backpack-integrated antenna prototype can form high-quality off-body communication channels.