1- Confirm the waterproofing of your shoes
Whether your shoe is warm or not, it will be of no use if you do not make sure it's waterproof. At Quechua the waterproof test is done on movement : the submerged shoe goes under 3 levels of waterproof flexure, which are as follows: 2000 flexures (around 2 hours of walking), 4000 flexures (around 4 hours of walking) and 8000 flexures (around 8 hours of walking). You can find this information in the technical characteristics of the products description and you will have a good idea of what your shoe can bear!
If the waterproofing of your shoe has faded over time, do not hesitate to revive them with a re-waterproofing spray.
And if you have good hiking shoes but they are not waterproof, you can use gaiters. They will cover and protect you from moisture from above (rain).