When you have a model trained to recognize images of size 128x128, you don't need to train another model from scratch to be able to recognize images of size 256x256.
What you can do instead, is: retrain the prevoius model for the new task.
Unfreeze the model, select a new learning rate about 5x-10x smaller than where loss starts to increase (rule of thumb). Then, run a training epoch and observe resulting loss.
Also, regarding identifying the learning rate to use - Jeremy noted he could imagine picking a learning rate being done automatically as a result of a project that maybe you could take up! He said he was sure it's possible to do it.
Source: Lesson 3: Deep Learning 2019 - Data blocks; Multi-label classification; Segmentation