Fine-tuning speech representation models can enhance performance on specific tasks but often compromises their cross-task generalization ability. This degradation is often caused by excessive changes in the representations, making it difficult to retain information learned during pre-training. Existing approaches, such as regularizing weight changes during fine-tuning, may fail to maintain sufficiently high feature similarity with the pre-trained model, and thus could possibly lose cross-task generalization. To address this issue, we propose Speech-FT, a novel two-stage fine-tuning framework designed to maintain cross-task generalization while benefiting from fine-tuning. Speech-FT first applies fine-tuning specifically designed to reduce representational drift, followed by weight-space interpolation with the pre-trained model to restore cross-task generalization. Extensive experiments on HuBERT, wav2vec 2.0, DeCoAR 2.0, and WavLM Base+ demonstrate that Speech-FT consistently improves performance across a wide range of supervised, unsupervised, and multitask fine-tuning scenarios. Moreover, Speech-FT achieves superior cross-task generalization compared to fine-tuning baselines that explicitly constrain weight changes, such as weight-space regularization and LoRA fine-tuning. Our analysis reveals that Speech-FT maintains higher feature similarity to the pre-trained model compared to alternative strategies, despite allowing larger weight-space updates. Notably, Speech-FT achieves significant improvements on the SUPERB benchmark. For example, when fine-tuning HuBERT on automatic speech recognition, Speech-FT is able to reduce phone error rate from 5.17% to 3.94%, lower word error rate from 6.38% to 5.75%, and increase speaker identification accuracy from 81.86% to 84.11%. Speech-FT provides a simple yet powerful solution for further refining speech representation models after pre-training.