If a Futures contract has two separate dates $T_d$ (when deliveries may begin) and $T_f$ (which is the last date by which delivery must take place) then you cannot use the formula found in introductory books such as Hull:
For example for Bond Futures $T_d$ is about one month before $T_f$. In that case which value of T would you use? The formula has been derived under the assumption of a single delivery date (which is true for example for Stock Index Futures).
For such cases the choice of the delivery date itself becomes a variable in the problem, and there are various (complicated) theories and methods available (generally described as Delivery Option analysis). Markets participants are certainly aware that the delivery date is uncertain for these kinds of futures; some (such as myself) simply avoid trading the future between $T_d$ and $T_f$ when the delivery may take place, and leave the "delivery timing game" to the specialists. Between time 0 and $T_d$ they assume delivery will take place at whichever of $T_d$ or $T_f$ is least favorable for their position. Others use much more complex optimization models.
But yes, real world futures can be more challenging that those presented in introductory courses.