Automatic speaker verification is susceptible to various manipulations and spoofing, such as text-to-speech (TTS) synthesis, voice conversion (VC), replay, tampering, and so on. In this paper, we consider a new spoofing scenario called "Partial Spoof" (PS) in which synthesized or transformed audio segments are embedded into a bona fide speech utterance. While existing countermeasures (CMs) can detect fully spoofed utterances, there is a need for their adaptation or extension to the PS scenario to detect utterances in which only a part of the audio signal is generated and hence only a fraction of an utterance is spoofed. For improved explainability, such new CMs should ideally also be able to detect such short spoofed segments. Our previous study introduced the first version of a speech database suitable for training CMs for the PS scenario and showed that, although it is possible to train CMs to execute the two types of detection described above, there is much room for improvement. In this paper we propose various improvements to construct a significantly more accurate CM that can detect short generated spoofed audio segments at finer temporal resolutions. First, we introduce newly proposed self-supervised pre-trained models as enhanced feature extractors. Second, we extend the PartialSpoof database by adding segment labels for various temporal resolutions, ranging from 20 ms to 640 ms. Third, we propose a new CM and training strategies that enable the simultaneous use of the utterance-level and segment-level labels at different temporal resolutions. We also show that the proposed CM is capable of detecting spoofing at the utterance level with low error rates, not only in the PS scenario but also in a related logical access (LA) scenario. The equal error rates of utterance-level detection on the PartialSpoof and the ASVspoof 2019 LA database were 0.47% and 0.59%, respectively.