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Calculations of molecular spectral properties, like photodissociation rates and absorption bands, rely on knowledge of the excited state energies of the molecule of interest. Protocols based on the variational quantum eigensolver (VQE) are promising candidates to calculate such energies on emerging noisy intermediate-scale quantum (NISQ) computers. The successful implementation of these protocols on NISQ computers, relies on ansätze that can accurately approximate the molecular states and that can be implemented by shallow quantum circuits. We introduce the excited qubit-excitation-based adaptive (e-QEB-ADAPT)-VQE protocol to calculate molecular-excited-state energies. The e-QEB-ADAPT-VQE constructs efficient problem-tailored ansätze by iteratively appending evolutions of qubit excitation operators. The e-QEB-ADAPT-VQE also improves on previous ADAPT-VQE protocol in that it is designed to be independent on the choice of initial reference state. We perform classical numerical simulations for LiH and BeH2 to benchmark the performance of the e-QEB-ADAPT-VQE. We demonstrate that the e-QEB-ADAPT-VQE can construct highly accurate ansätze that require at least an order of magnitude fewer cnot gates than standard fixed unitary coupled-cluster ansätze, such as the UCCSD and the GUCCSD. We also show that the e-QEB-ADAPT-VQE is more successful in constructing ansätze for excited molecular states than other ADAPT-VQE protocols.