Since we do not have all of the facts pertaining to your requirement, solicitation/contract or contractor(s), the following answer is based solely on the background and question provided. As we do not have access to the contract folder or particulars that apply to this situation, we highly recommend you consult the Contracting Officer and possibly the Legal Office.
Based on your question background, you have already determined that the new work has been determined to be within scope of the original contract; therefore, this response will not address the issue of scope as you have already confirmed the added CLIN requirement is within scope. Concerning adding a new CLIN via the modification, refer to DFARS 204.7106. In accordance with DFARS 204.7106, for contract modifications, if NEW ITEMS ARE
ADDED, assign new contract line or subline item numbers or exhibit line item numbers in accordance with the procedures established at DFARS 204.7103, 204.7104 and 204.7105. DFARS 204.7103-1 specifically addresses the need for a CLIN to have four characteristics including the one you discussed in your question background - separate delivery schedule, period of performance or completion date.
You will also need to confront another matter. Be careful that you do not attempt to "reobligate" funds for a new purpose (CLIN 0003) after the fiscal period of availability from funds that will not be liquidated for the purpose against which they were obligated (CLIN 0001). This could violate the bona fide need of the original fiscal period of availability. Further, if the purpose of CLIN 0003 is to extend the period of performance of the services acquired under CLIN 0001, this would result in a period of performance of more than one year in duration, which is not allowable when buying severable services, unless you use options for extensions.