If your booting into your old linux mint version instead of the new one you installed you need to access the boot loader and select the version you want to start up. here's how you do that
1) if the computer is running, shut it down first
2) Once it's shut down, press the on button to start it up again
3) As soon as the bios starts to load press and hold the "Esc" button on the top left of your keyboard. this will force the boot loader to start up
4) As soon as the boot loader starts you will have a screen that shows you all of the OS versions installed on the computer. The one on the very top is normally the default OS to boot to.
You only have a few seconds to make a selection before the default OS boots. To avoid this as soon as the boot loader appears on your screen press the shift or space key (either). What this does is cancel the boot timer and waits for yo u to make a selection from the screen of the OS you want to boot to.
for instance, if you old Linux Mint OS was 13 it will be at the top of the boot loader screen. I'm assuming you upgraded to Linux Mint 17 (or what ever version you just upgraded to). Look down the list of choices until you see the new version you installed and select it, then hit enter.
Your computer will now boot to that version instead of the old version.
Hint/trick - If you want to monitor/see what's happening while the system is booting press the "F2" key on your keyboard and a step by step of what's happening will show on your screen until the OS is loaded and you are looking at the sign-in screen.
On the older versions of Linux Mint when you had more than one version or OS installed at the same time the boot loader would always be the screen you were presented with so you could choose which version to boot too. That changed on the newer Linux Mint versions and the boot loader screen is bypassed unless you hit and hold the "Esc" key as soon as the computer starts to power up.
Once you choose the version you want to boot to from the boot loader screen it should automatically choose that same version every time you restart your computer. If it should boot into the older version, just restart and hold the "Esc" key and select the version you want to boot to