You can easily fix a fishing rod without repairing it in an anvil by using this trick. It doesn't work with enchantments but it is still a glitch.
HOW TO REPLICATE
1. Take a fishing rod that is low on durability. In creative, you can do this by typing /give @s fishing_rod{Damage:30}.
2. Now, take a carrot and combine the two in a crafting table. Take out the resulting carrot on a stick.
3. Take the carrot on a stick and find a pig.
4. Equip the pig with a saddle by right-clicking and right-click again to ride it. Now, keep right clicking to slowly reduce the durability on the carrot on a stick.
5. When the carrot on a stick breaks, it transforms into a full-durability fishing rod, even though the rod you started with was lower than full.