This paper presents an extension of the L1 adaptive output feedback controller to systems of unknown relative degree in the presence of time-varying uncertainties without restricting the rate of their variation. As compared to earlier results in this direction, a new piece-wise continuous adaptive law is introduced along with the low-pass filtered control signal that allows for achieving arbitrarily close tracking of the input and the output signał of the reference system, the transfer function of which is not required to be strictly positive real (SPR). Stability of this reference system is proved using small-gain type argument. The performance bounds between the closed-loop reference system and the closed-loop L1 adaptive system can be rendered arbitrarily small by reducing the step size of integration. Simulations verify the theoretical findings.