You can always install any year and SP version that was released on or before the date your subscription expired.
You do not need your subscription to be active to install. During the install process, the install manager will check your old license number and validate it for the package you are trying to install. If that package was released AFTER your subscription expired it will fail - but if it was released before it will install without issues of the license.
The problem is having access to download the version you want if you don't have an active subscription.
I periodically download versions and SP when I am on subscription, even if I don't intend to install them, so I always have a backup.
Does anyone at your location have a current subscription? If yes, then they can do a "Download Only" and then give that to you to install.
If you have an old download of the version you want to use then that will work.
If you can not find a copy, contact your VAR. They should have old versions that they can give you (even if you are off subscription the VAR will usually support you with this).
For any time in the future that you or someone in your company is on subscription, be sure to download the latest version and SP available before your subscription expires.
Also, if you were on an earlier version (say 2015) (so the new version and SP were already released but your hadn't upgraded yet) Be sure to also download that years version and SP5. In case you have problems with 2016 you can go back to 2015 SP5 and install that.
EDIT: I also have found it is better to do a "Download Only" instead of installing directly from the website.
There are several reasons for this:
You download "all" the required files and prerequisites without interruption from the install process - less chance of download errors.
You have a "Full" copy of all the files you would need to later do an install / upgrade / repair.
The download can be done overnight or when you are not on your computer so it does not tie you up (downloads are big and take a while - especially on a slow network).
If you have a problem and the install does not successfully run, you don't have to go back online and download again.
If you later have a problem, you have the local files and can do a repair / reinstall without having to download again.
The install or repair is much faster if the files are local.
You have a backup or earlier SP if there are problems and you want to go to an earlier SP (you would not want to go to an earlier version (earlier year) if you have made or edited any parts, drawings or assemblies because they will not be readable by earlier versions - but can be with same version, earlier service packs)
Another thing you should do.
Since your license was associated with the old computer and you can not release that license (as you would usually do when changing computers), contact your VAR and have them contact SW to release that license. If you do not, it may say you already have your license associated with the old computer and it is not available.
Thank You about quick and detailed response!