How early I can apply for the renewal of an Indian passport from a new city?

Your own government's instructions indicate you can apply as soon as you have the required proof:

Change of Address

Q61: How do I change the address on my passport?
A: To change the address in the passport, you have to apply for a "Re-issue" of passport and get the specified change done in the personal particulars.

Table 3, found on page 11

Proof of Present Address.
For Proof of Address attach one of the following documents:

a. Water Bill
b. Telephone (landline or postpaid mobile bill)
c. Electricity bill
d. Income Tax Assessment Order
e. Election Commission Photo ID card
f. Gas connection bill
g. Certificate from Employer of reputed and widely known companies on letter 
   head (Only public limited companies can give address proof on company 
   letter head along with seal. Computerised print-outs shall not be 
   entertained)
h. Spouse’s passport copy (First and last page including family details 
   mentioning applicant's name as spouse of the passport holder), (provided 
   the applicant’s present address matches the address mentioned in the 
   spouse’s passport)
i. Parent’s passport copy, in case of minors (First and last page)
j. Aadhaar Letter/ Card (Aadhaar letter/card or the e-Aadhaar (an 
   electronically generated letter from the website of UIDAI), as the case 
   may be, will be accepted as Proof of Address (POA) and Proof of Photo- 
   Identity (POI) for availing passport related services. Acceptance of 
   Aadhaar as PoA and PoI would be subject to successful validation with 
   Aadhaar database.)
k. Rent Agreement

You have to give details of all places you have stayed during past one year and the police verification would be done at all those places.

It will be better if you can get the passport reissued and police verification done at current address itself.