I have faced a similar challenge is my organization, where we were hiring around 50 to 70 employees a week.
We segregated the work into -
High Cost and High Impact Work,
Low Cost and High Time Work.
Sourcing, Screening and Interviewing were classified as High Cost and High Impact Work, as when they are outsourced - the cost is high, also if done incorrectly they impact the business.
So they were done by the team. Also when the team's bandwidth used to get stretched - we used to outsource - low cost positions, and still keep the high cost position sourcing with us.
We were also able to save time by introducing screening criteria of Aptitude Tests, and Video Interview Pitches. Only those who scored above a particular threshold were taken to the Video Interview Round, and Later those whose communication skills were upto the mark in the Video Interview Round - were taken for the final round of interview - where there was in person investment of time.
The Data Entry, Coordination etc - where the learning is quick, and is non value added - was done by Contract Employees - who were relatively low cost resources.
Alternatively depending on your budget - there are softwares which can automate these entire process, and also Consultants to whom you can outsource the entire process.
Your approach will depend on the budgets that you have.