We report on the experimental study of the photoconductive antennas (PCAs) - detectors based on superlattice heterostructures (SLS) InGaAs/InAs/InAlAs with different types of elastic stresses in their functional layers. By using our laboratory time-domain THz spectrometer we measured and compared the detected THz signals, noise characteristics, and signal-to-noise ratios of the developed bow-tie PCA-detectors at different average probe power. We showed that the SLS-based PCA-detector with elastic stresses of both compression and tension in the layers demonstrates increased THz detection bandwidth compared to that for the SLS-based PCA-detector with only compression stresses in the SLS layers in the whole range of the optical probe power. We thus demonstrate that modification of the SLS via introduction of the elastic strain in the crystalline lattice of its layers could become an efficient approach to enhance the PCA-detectors performance leading to implementation of these PCAs to the spectroscopic THz setups.