We consider the linear delay dynamic equation (*) x Δ ( t ) + p ( t ) x ( τ ( t ) ) = 0 for t ∈ [ t 0 , ∞ ) T . We enhanced the results in Wu and Zhou [2007. Stability for first order delay dynamic equations on time scales. Computers and Mathematics with Applications, 53(12), 1820–1831] by removing the boundedness criterion from the delay and graininess functions to cover the result in Zhang et al. [1999. Global attractivity of difference equations with variable delay. Dynamics of Continuous, Discrete and Impulsive Systems, 6(3), 307–317]. Furthermore, we have defined new functions that will assist us in proving our main results. We have presented some examples that illustrate the essence of our results in this work.