We recently started using some free hiring software called iKrut. In the jargon, I think it would be called a free applicant tracking system or free ATS.
I used to work for a staffing firm, and while it would probably work well for smaller ones, I would guess it's not ideal for bigger firms because it lacks the client record side of things, i.e., logging calls to them. However, since I now work for a direct employer, it does exactly what we need it to do and has transformed how I recruit. The fact that we don't pay anything makes it even better. Exactly like you, I used to rely on resumes coming in via email, swamping my inbox, then logging them into Excel to track where they've come from and what stage they were at... and this was so time-consuming. I think Zoho offers something similar but I'm not sure.