Currently there is no way to do this. This is because Test Complete is being launched under the Jenkins Job process, a new job will be a new process so there is no way to persist Test Complete between jobs.
However, a couple points I would like to make, Test Execute would be a better fit for this scenario as it does not need to launch the full IDE, just the execution portion which is much much faster.
Another thing, if your tests are failing, I highly doubt it has anything to do with the time it takes to open / close Test Complete. That is outside the scope of your tests running. If your tests are failing there is most likely some other reason which needs to be investigated.
One solution would be to only copy over your project files and not the entire suite so that when each project is opened it's not opening the entire suite of however many projects all at once. Project suites are nice for development, but do not really provide much benefit in terms of running tests, especially since the more individual projects you add to it, the slower it will be to open and the more resources it will consume.
