Know the feeling
I am the Network/Systems/Helpdesk personel at my company for roughly 75 desktops, 25 servers, and all the network equipment. Our management has also been hesitant to hire the additional staff, but they have been very understanding when it comes to how that choice affects response time.
The other item I would think to consider is how much change the environment needs to undergo. Our environment has 'calmed' significantly over time, making it far easier to run with just one person. Try not to staff up just to settle some immediate issues and then have little for people to work on afterwards. Consider one full time admin and possibly a part-time or intern to assist with the daily fire fighting until things become stable.
