Depends on the nature of your contact with your old provider and their policies/terms and conditions.
E.g. if it's a bill pay contract that still has time left to run then you will have to contact then.
Probably best to contact them just to make sure that you don't owe them anything and will not continue to be charged.
Not sure why you mention the PUK as that has nothing to do with moving providers.
Hope this helps.