Imaging products like Acronis have a couple of capabilities:
1: making and saving an image of a drive, usually to another hard drive. Then "restoring" that image if and when the original hard drive installation has major issues. Think of this as making a copy of your system and later placing that copy back on a hard drive. This is usually done to recover from disasters, like a failed hard drive or accidental deletion of important files.
2: "Cloning". This is copying a system on one hard drive to another hard drive in real time. You don't save an image and later restore it. This is normally done when things are working well, but you just want to move to a faster or larger drive.
If I understand you correctly, you would want to use the second method. Acronis can do that and I think the other standard imaging applications can do it as well.
In your case, you would connect the new drive and then "clone" from the old drive to the new drive.