Using a cold sourdough starter

I’ve just fed my sourdough starter (was kept in the fridge) with a plan to make English muffins tomorrow. Usually if I feed it in the morning, I’ll give it another feed in the evening and make the dough in the morning, otherwise I just give it one feed in the evening with the same plan to make the dough in the morning.

I’ve seen where people put their risen starter in the fridge for whenever they are ready to bake, and since I refreshed my sourdough with more than enough starter for my bake, my plan is to pop it in the fridge tonight and make my dough in the morning. My dough then goes straight to the fridge for 24 hours for bulk ferment. Then I shape and let rise, and cook on a griddle. 

My question is, since I’m starting with a cold sourdough starter to make my dough, will I need to make any adjustments, for instance instead of putting directly into fridge after mixing, would I need to let it sit out a few hours first? Will starting with a cold starter change the flavor, making it more sour?

Thanks for any insight.