raw copy of 1TB takes 22 hours at constant speed of 100MBit/s https://calculators.io/bandwidth/ Depending on your hardware and bandwidth 2 days sounds not really bad. I would expect big files like archives or media to transfer at maximum speed. small files like text or thumbnails are much slower.
If you have physical access to both hosts you could ship the data by external USB drive - this may happen faster… depending on the kind of your data this could take long time as well and you need to copy the data twice (from original host to the drive and from the drive to destination host). If you got the data replicated to you local client you could upload this to new host from there - again subject to hardware and bandwidth limits.
It depends on the type of data. If these are all zip archives, then you won’t gain anything, if there are a lot of text files, then yes. In the linked question, they mention an article with tests on mixed data and the increase of the effective bandwidth due to compression.