A tightly interlaced matched ambidextrous bilayer (TIMAB) is a sculptured thin film consisting of a cascade of one-dimensional cells made up of two layers of chiral constituents that are identical except for their structural handedness and are as thick as their structural period. Periodic TIMABs, where the structural period of the chiral constituents is the same in all bilayer cells, exhibit a narrow polarization-universal bandgap on axial excitation. We propose linearly chirped TIMABs, in which the structural period of the chiral constituents increases with the position of the cell in the structure and show that they exhibit a significantly wider bandgap (more than four times) than the periodic TIMABs.