A customer deposited a check yesterday morning and we received information today that the check may not be paid. Can still place an exception hold or does it have to be the day of the deposit?

Yes, you can still place an exception hold, as long as you meet the notice requirement:
(ii) Timing of notice. (A) The notice shall be provided to the depositor at the time of the deposit, unless the deposit is not made in person to an employee of the depositary bank, or, if the facts upon which a determination to invoke one of the exceptions in paragraphs (b) through (f) of this section to delay a deposit only become known to the depositary bank after the time of the deposit. If the notice is not given at the time of the deposit, the depositary bank shall mail or deliver the notice to the customer as soon as practicable, but no later than the first business day following the day the facts become known to the depositary bank, or the deposit is made, whichever is later.

Assuming that you will be placing that hold to delay availability to the seventh business day, don't forget that you base your count on the date of the deposit, not the date on which you received the information about the questionable status of the check.