You can claim an exotic from the previous season. If you got the anual pass from humble bundle.

[DJukor]

https://www.bungie.net/7/en/Seasons/PreviousSeason
Unfortunately you cant progress the older season.

There is also a plugin called Season Pass Pass that lets you get even older season exotics such as Defiance and Deep.

[trukr]

But you can play the seasonal stories from the three past seasons and use the vendors.

[DJukor]

Originally posted by trukr:

But you can play the seasonal stories from the three past seasons and use the vendors.

You can but the 3 older exotics are season pass track only.

A Ice bow a crystal Grande Loucher and a Rifle i am gonna claim later when i rise my light level. As the gear will be your - artifact light level when you claim it.