I want to share some facts with you
1. Puppy needs first shot of rabies vaccine
at age of 3 months, earlier than 12 weeks vaccine may not be effective due to interference by antibodies
transferred by mother.
2. Second shot should be given at age of 12 to 16 months.
3. Boosters doses should be given at every 1-3 years as per laws of country.
4. It takes at least 14 days to develop antibodies against rabies after shot of vaccine
Now I come to your question
1.You have not made it clear that when the first dose given to Puppy if it was given before 3 months of age then it was ineffective.
2.Latest dose given only 10 days back and to develop it needs 14 days for 100% protection
So in short I want to say that if the first dose was given to Puppy before the age of 3 months than you SHOULD VACCINATE your chid.
Wishing good health for your family.