2 administrators are already a team, and given the distributed nature of the process, the asymptotics of the management task is 10x3x2 = 60, this is essential for management (10 servers, 3 is the average complexity factor since the server management is divided into equipment, OS and application systems, 2 administrators).
Therefore, the tool must be appropriate - management tasks or tickets. For any work, administrators should create a ticket (task) with a description, agreed terms, specified infrastructure objects (servers, with details - equipment, OS, application systems, etc.). There must be an execution schedule that includes documentation (which was actually done by the contractor).
By implementing such a system, you will receive detailed logging of what was planned and when, how and by whom it was executed.
The help desk system (for example, Spiceworks , but a lot of options in general) or the project management system (for example, Asana or Atlassian Jira , but the options are dark again) are suitable as tools. Old Mantis can not help but remember a good memory. The choice will depend on what you like more and whether you are ready to spend some money. IMHO so it's better to take an inexpensive Cloud-system so as not to bathe with local installation and administration, the budget of $ 10 -20 $ per month for the admin can be thought up ;-)
And most importantly, a base for infrastructure growth will be created, after some time you can safely go to the authorities with a proposal to purchase 10 more servers and hire another administrator, as the efficiency and transparency of the processes will definitely increase ;-)