Katy, as Adam mentions above, this is a feature request that is in our backlog of future enhancements. In the meantime, the workaround we've been suggesting is to record a message that begins with an introductory repeating message asking the customer to hold for an upcoming message that lasts for a few seconds followed by the actual message and even possibly repeating that message a 2nd time. That way, even if the beep or greeting blocks some of the beginning, the message itself can still be heard. Something like this...
"Please wait for an important message...please wait for an important message...Hello, this message is for Chad McCormick from Canyon Point Dental. We are calling to remind you of an appointment on 4/12 at 3:30pm. Please call us if you need to reschedule...Again, this message is for Chad McCormick from Canyon Point Dental. We are calling to remind you of an appointment on 4/12 at 3:30pm. Please call us if you need to reschedule. Thank you, goodbye"
Hope this helps in some way!