It seems noble to solve a problem outside of your sphere of responsibility. If you’re solving this problem because there is no permanent responsible party, you may have just inherited a new task. Management won’t solve a problem that doesn’t cross their radar. If you keep handling the task, it never will. Identify the problem(s) and express the need for additional resources or a more creative solution.
Don’t silently solve additional problems while your own work stacks up. You could become overwhelmed and eventually burn out.
I value efficiency and strive to improve productivity. In the past I’ve used the passive tracking service RescueTime, but I’ve found that I don’t review my stats very often. I started tracking with Toggl a few weeks ago. I’ve found that right from the start, being forced to give the task a label and project makes you aware of the type of work you’re about to do. If it’s not in line with your values, principles and goals, you tend to rethink the task. This is huge! In the past I’d go through email in the morning and fall into the busy trap, working on things as I come across them instead of processing everything that takes over two minutes. Getting the right mindset before you begin work instead of just working ensures you are meeting goals and helps to pinpoint inefficiencies. It’s really a very short cycle of Beginning with the End in Mind. Very effective way to make sure you’re working on the right thing!