How to make my follower to wear same type of armor as her default, but improved?

The mod you linked to, TCG - Tradable Companion Gear can indeed allow you to remove all of your follower's equipment, even those that are considered part of your follower's "default gear" (doesn't appear in the follower's inventory, and is unremovable in the vanilla game) - the Blades armor, in this case.

TCG is now also part of a bigger follower ovehaul mod, UFO - Ultimate Follower Overhaul. They are made by the same author.

TCG description from its webpage:

• All followers gear can be traded.
• Default hidden bow and arrows are tradable and not respawnable.
• Followers will not revert to default outfits or unequip gear even if dismissed.
• Followers will wear what is placed in their inventory.

With TCG or UFO, just open up the item inventory screen of Lydia and remove her default Blades armor, and then put in the improved Blades armor that you have.

IF you are playing on the PC, the following steps should work:

• Open the console
• Click on her face
• Type removeallitems and press Enter