can I connect the new earth with the existing earthing system?
You can add additional grounding rods to an existing grounding system. Grounding rods on one side of the building can be bonded to grounding rods on the other side of the building using a ring ground or ground ring (same thing with different name), which is a bare copper wire strung along the foundation either slightly below the top of the soil, or sometimes a short distance above the top of the soil.
Although you can add additional grounding rods, what you should NOT do is connect any part of the grounding system to the neutral wire at any point other than in the principle breaker panel. Such a practice will generally degrade safety, and is a code violation in the US and may be in your jurisdiction as well.
Will it improve the overall safety?
If you live in an area with frequent lightning strikes, adding a ground ring around the building and multiple grounding rods may improve your grounding system's ability to absorb a nearby strike.
If your soil is very dry, or you live in a rocky area, and as a result your ground resistance is high, say over 25 \$\Omega\$, then such an addition may improve your lightning protection. However, the electrical code in your jurisdiction may already require that your grounding system's resistance to ground be below a given level, such as 25 \$\Omega\$. In such a case, multiple ground rods may already be present.
However, as far as protection from faults in your electrical system, adding a ground ring, or extra ground rods will likely not improve your safety. Ground Fault Current Interrupters (GFCIs) or their equivalent in your country, probably provide the best safety from shock and electrocution due to faults in your electrical system. The marginal improvement in safety from multiple ground rods, except in the cases mentioned above, will be minimal by comparison.
I am not a licensed electrician, and I am certainly not a licensed electrician in your jurisdiction. You should take the advice of such a licensed electrician, rather than from a stranger on the internet. However, my personal opinion is that 1) it won't hurt to add extra grounding rods, but 2) it is not worth adding extra ground rods except when either one rod provides insufficiently low resistance to earth, or if you live in an area where lightning strikes are frequent or a special concern. Again, this is the opinion of someone who is, to you, a random person on the internet. Treat my opinion as something you need to verify.