You have a big problem my friend. Intake is really the only solution for you. I guess you're not using a graphics card. It is still problematic, as the airflow is very convoluted. The intake and the exhaust are basically facing the same way.
Would you like to use cardboard? There are ways to improve the flow, but some creativity is required. A new case is really called for, but you can create ducting to shape the airflow. Some double-sided tape will keep it in place till you upgrade to a new case.
Ensure you leave a bit of airflow for the drives toward the front. Split the flow into two channels, and direct it upward to the CPU. The PSU is your only exhaust route this way.
If you're using it as an exhaust then divide the case into two portions, one for the airflow across the lower half of the motherboard. The PSU then has to take the entire cooling load of the CPU. A good way to ensure this is effective is to remove the PSU top cover. This creates a lot of EMI within the case, but increases the cooling effectiveness of the puny little PS exhaust fan. I take your load is not very demanding, so the PSU should work fine with the residual airflow.
EMI would be concern if you're running audio or instrumentation apps, but I'm assuming you're not into all that. I'm assuming a lot of things, one of them being that you're not prepared to change the case. That, of course, would probably be the best solution.
Anyway try both intake and exhaust and see which gives better temps. Look for a tradeoff between CPU temp, case temp and HDD temp. In the ideal case let the system run for about two hours with your normal load. Post back temps in both configs.