Is there any role of a design engineer when a structure is already constructed?

The main role of design engineer is to design a structure.
During the construction stage the engineer needs to make sure that required design standards and specifications are met.
Apart from that when the structure is already built and it needs extension it needs the designer involvement once again as he is the one keeping records of design calculations and drawings and can decide on possibility of extension for example for construction of additional floor.
From maintenance point of view I don't think it needs much involvement of design engineer but any modification in structure or its use other then considered in design must be consulted with the design engineer.

yes there may be changes required in the design at later stages too

Yes