Hi Aidan,
If you consider your first set of images - you must have performed a StarAlignment during that process, and you must have used a referece image against which you aligned all other images.
If so, then what you now need to do is to StarAlign all of your new images against the same (original) reference image.
At least all of your data is now correctly aligned.
You could now proceed with ImageIntegration of your new images - if you did this you would then have two 'MasterImages', and you could simply run both of these through a 'standard' (all parameters reset) ImageIntegration Process to generate a final MasterImage. The alternative - though pretty much identical - method (here) would be to add both of the images together using PixelMath.
Of course, if you still had all of your original calibrated Lights, then you could run that set, and your new set - together - through a single ImageIntegration Process. You might find that this gives you the best noise rejection (and hence the greatest increase in SNR).
You should consider experimenting (or save the task for a rainy day!).