We have tried to patent a system few years ago, but we couldn't patent it because we had sold few systems more than 1 year before we tried to obtain the patent. Now we have modified the system in a way that it is much better improved. Can we patent the complete system with the improvement now? The system remains basically the same, but now we can apply it to other industrial processes for having improved the temperature that it supports. Thank you.
Yes, you might be able to maintain the position that the improvement is patentable, as something new and unobvious over the original version, and you might get a patent for it.
But bear in mind that such a patent won't cover what was already in the prior art and free-to-use. Whether a patent claim is legally valid, and whether it is commercially valuable are two quite separate questions with potentially different answers.
One crucial question would be, does the improvement give the improved product a competitive edge, would the original in the hands of a competitor still amount to effective competition with the new version? If the market doesn't care which version it gets, a patent on the improvement might possibly not be worth having, patentable or not.
You might be able to. Many patents would be considered improvements on something that came earlier. If the newer system is structurally different and that difference is non-obvious from everything that preceded it, including your previous system.
Depending on the details you can patent both the improvement and the entire improved system. Also, the new use - as a method - may be patentable.
In your example, the original system was your own and wasn't patented. In a different case where the older system might have been patented by another a commercial issue comes up. It might be that the improvement is such that any system incorporating it might inherently infringe on an existing patent relating to the older system. In that case anyone practising your patent would need a license to both your patent and the earlier patent.