Open Source Software (OSS) sustainability relies on newcomers transitioning to core contributors, but this pipeline is broken, with most newcomers becoming inactive after initial contributions. Open Source Software for Social Good (OSS4SG) projects, which prioritize societal impact as their primary mission, may be associated with different newcomer-to-core transition outcomes than conventional OSS projects. We compared 375 projects (190 OSS4SG, 185 OSS), analyzing 92,721 contributors and 3.5 million commits. OSS4SG projects retain contributors at 2.2X higher rates and contributors have 19.6% higher probability of achieving core status. Early broad project exploration predicts core achievement (22.2% importance); conventional OSS concentrates on one dominant pathway (61.62% of transitions) while OSS4SG provides multiple pathways. Contrary to intuition, contributors who invest time learning the project before intensifying their contributions (Late Spike pattern) achieve core status 2.4-2.9X faster (21 weeks) than those who contribute intensively from day one (Early Spike pattern, 51-60 weeks). OSS4SG supports two effective temporal patterns while only Late Spike achieves fastest time-to-core in conventional OSS. Our findings suggest that finding a project aligned with personal values and taking time to understand the codebase before major contributions are key strategies for achieving core status. Our findings show that project mission is associated with measurably different environments for newcomer-to-core transitions and provide evidence-based guidance for newcomers and maintainers.