As you asking for the Govt jobs after completing the Engineering from the ECE ,you can get into various fields in public sector.

PSU(Core field)
Telecom field
Defense sector
Banking sector
Railway sector
Administrative sector.

For getting jobs in PSUs like
NTPC,
BHEL,
IOCL
,ONGC,S
AIL and COAL INDIA
,you have to clear GATE exam with a good score.
For other PSUs like
BEL
,Oil India Limited,
GAIL (India) Limited,
Bharat Dynamics Limited,
Bharat Petroleum Corporation Limited,and other miniratna and navratna companies ,you have to clear the written test conducted by the respective company.

For getting job in
Telecom Units of public sector like Bharat Sanchar Nigam Limited(BSNL),Mahanagar Telephone Nigam Limited(MTNL),you have to clear the written exam conducted by the respective units like JTO in BSNL.

For getting into defence
sector, you have to clear CDS exam.
For getting jobs in Nationalized Banks as Clerks and Officers ,you have to give IBPS exam for assistants(clerks)and po.
For getting jobs in Railways in different posts like
Junior Engineer
,Section Engineer,
Station Master,
you have to clear IRS and RRB exams.

you may also apply for the IES Examination conducted by the UPSC every year for the Engineers