Modern applications of Internet of Things (IoT) devices require cheap and effective methods of measurement of physical quantities. Cheap IoT devices with sensor functionalities can detect a lack or excess of substances in everyday life or industry processes. One possible use of tension sensors in IoT applications is the automated replenishment process of fast moving consumer goods (FMCG) on shop shelves or home retail automation that allows for quick ordering of FMCG, where the IoT system is a part of smart packaging. For those reasons, a growing demand for cheap and tiny tension sensors has arisen. In this article, we propose a solution of a small flexible tension sensor fabricated in an amorphous InGaZnO (a-IGZO) thin-film process that can be integrated with other devices, e.g., near-field communications (NFC) or a barcode radio frequency identification (RFID) tag. The sensor was designed to magnify the slight internal changes in material properties caused by mechanical stress. These changes affect the dynamic electrical properties of specially designed inverters for a pair of ring oscillators, in which the frequencies become stress-dependent. In the article, we discuss and explain the approach to the optimum design of a ring oscillator that manifests the highest sensitivity to mechanical stress.