What's the difference?
I did a little research, and this article does a good job at summarizing both the taste difference, and the growing difference.
Young ginger (or spring ginger) is harvested at the start of the growing season, before it has gotten as fibrous, and when the exterior skin is still thin & delicate. It also has a more mild flavor. Mature ginger (what the OP calls "old ginger" is simply grown longer and has a tougher skin, and more pungent taste.
Full text from the above article:
Fresh ginger is available both young and mature.
Spring ginger. Young ginger which is sometimes called spring ginger has a very thin skin that requires no peeling. The skin is edible. Young ginger is very tender and has a milder flavor than mature ginger. Young ginger is found in Asian markets in the springtime.
Mature ginger. Mature ginger is usually harvested in the fall and has a tough skin that must be carefully peeled to reach the flesh underneath. Mature ginger is more pungent than young ginger.
Just harvested ginger—whether young or mature–should have a fresh, spicy fragrance.
So what can you do?
Once it's harvested, there isn't really a way to "mature" it at home in your kitchen. That needs to happen when it's still in the garden. You could try simply using a "heavier hand" and use more ginger to compensate for the slightly more delicate flavor, though that still won't be the same thing as using mature ginger to begin with.