3 Phase DYn1 transformer. Can it be connected externally DYn3 ? 1

You'll have to swap two H connections and two X connections. There are at least 3 combinations that will get you what you need. I'll let you have a bit of transformer vector fun; then pick the one that causes the least trouble when reconnecting.

I’ll see your silver lining and raise you two black clouds. - Protection Operations

Thanks Davidbeach. Vector realignment is a complex subject for me, especially when I have to swap terminals, but I will certainly try.
Meanwhile it seems that I made a mistake in stating the problem. I am describing below more accurately.
We have a 3 phase, 26.4 kV incoming from utility in the substation. There is a Dd0 transformer stepping 26kV down to 480 volt. Another DYn1 transformer 150KVA, steps down 480 volt to 120/208 volt for lighting/heating etc. This 120/208 supply is given to 3 pole Auto Transfer Switch, neutral being common with other incoming supply to ATS. The other low voltage supply of 3 phase 4 wire 120/208 volt, is coming to ATS from utility.
Let us say that ATS's first (Normal) supply, which is internally generated in the substation, is A, B, C and N.
The ATS's second (Emergency) supply is coming from utility, is A', B', and C' and N'. The N and N' go to common neutral bar of ATS.
I measured voltages to check if the 2 sources in ATS are in sync. They are not. The voltage measured are A-A' = 120 volt, A-B' = 240 volt and A-C' = 120 volt. (Similarly for B and C phases.) So there is a phase difference of 60 degree between the two sources in the ATS. Both sources have correct phasor rotation. Obviously, We cannot do anything to the 120/208 volt coming from utility.
I wonder if our DYn1 transformer can be externally rewired as DYn3, (which gives additional 60 degree shift) and in that case, will it bring both sources in sync in the ATS ? Thanks.

Any delta-wye transformer can be connected to provide a phase shift equal to any odd multiple of 30 degrees.

The one time I had to put that into practice the actual set of swaps was determined by swap that worked at the vertical dead end a span away from the horizontal dead end over the transformer. Once that was determined there was one swap of low-side cables that completed the desired phase shift.

I’ll see your silver lining and raise you two black clouds. - Protection Operations

You would have to disconnect all 3 coils at neutral ends. At this power, these should be welded.

No. All done external to the transformer. Draw up a Dy transformer, the terminals are H1, H2, H3 and X0, X1, X2, X3. Assign A to H1 and a to X1, B to H2 and b to X2, and C to H3 and c to X3. Work out the vector relationship of ABC to abc. Now swap things around so that you have B-H1, A-H2, C-H3, b-X1, a-X2, and c-X3. Once again, work out the vector relationship of ABC to abc. It's different and it's some odd multiple of 60 degrees different.

As a hint, there's a FAQ for this forum that describes the "conversion" between Dy1 and Dy11, also a 60 degree shift. That doesn't require any changes inside the transformer either.

I’ll see your silver lining and raise you two black clouds. - Protection Operations

Davidbeach, Thank you.
I was able to draw vectors and understand at least one cable reconnection configuration, which will give 60 degree shift. I did not attempt the other 2 combinations. But now I am convinced that it can be done and there are more than 1 ways to reconnect cables on primary and secondary to get the 60 degree shift.
Just a question of interest.......If the above 2 sources in ATS; one from utility 120/208V 3 phase 4 wire and second 120/208V 3 phase 4 wire from secondary of our Dyn1 transformer; were to be 30 degree apart, then reconnecting DYn1 would not be possible to give 30 degree shift such that both sources become in sync. Correct ?