Perfectly possible to migrate your current installation to a new hard drive be it SSD, mechanical or hybrid but you will need software to do it.
As Ian pointed out some hard disk manufacturers supply free software to do this task and the software is usually a re-badged version of Acronis True Image or a variation of.
I've used Acronis True Image for years now and have cloned dozens of hard disks with it, no problems.
I'm using a small 120Gb Samsung SSD drive atm and I keep all my data and games, including my Steam directory, on seperate storage disks, all of which are mechanical drives.
How big is your current Steam directory?