Fast execution of contact-rich manipulation is critical for practical deployment, yet providing fast demonstrations for imitation learning (IL) remains challenging: humans cannot demonstrate at high speed, and naively accelerating demonstrations alters contact dynamics and induces large tracking errors. We present a method to autonomously refine time-accelerated demonstrations by repurposing Iterative Reference Learning Control (IRLC) to iteratively update the reference trajectory from observed tracking errors. However, applying IRLC directly at high speed tends to produce larger early-iteration errors and less stable transients. To address this issue, we propose Incremental Iterative Reference Learning Control (I2RLC), which gradually increases the speed while updating the reference, yielding high-fidelity trajectories. We validate on real-robot whiteboard erasing and peg-in-hole tasks using a teleoperation setup with a compliance-controlled follower and a 3D-printed haptic leader. Both IRLC and I2RLC achieve up to 10x faster demonstrations with reduced tracking error; moreover, I2RLC improves spatial similarity to the original trajectories by 22.5% on average over IRLC across three tasks and multiple speeds (3x-10x). We then use the refined trajectories to train IL policies; the resulting policies execute faster than the demonstrations and achieve 100% success rates in the peg-in-hole task at both seen and unseen positions, with I2RLC-trained policies exhibiting lower contact forces than those trained on IRLC-refined demonstrations. These results indicate that gradual speed scheduling coupled with reference adaptation provides a practical path to fast, contact-rich IL.