This paper investigates a problem of stability analysis for a class of nonlinear systems with a time-varying delay taking both large and small values in an alternating manner. This problem is a delay-dependent stability analysis one but it shares some features with delay-independent ones. Hence traditional stability analysis techniques for systems with delay do not apply. By building upon our previous works, we first introduce the concept of delay-hybrid-dependent stability, which grasps the features of the delays described above. To ease the problem we investigate, we represent the studied system as a system with a switched delay. Then by using switching techniques and Lyapunov-Krasovskii functionals (LKFs), we provide a new stability criterion. Next, in the linear context, we show how the needed LKFs can be constructed by developing an original design of LKFs.