I play through Encore mode from Hydrocity all the way through to Titanic Monarch. I also find all special stage rings in these zones. Well, I believe I've found them all. If I've missed any, then please let me know where they are!

#SonicManiaPlus
#SpecialStageRings
#EncoreMode


Special Stage RIng Locations:

HC act1 #1 1:00
HC act1 #2 2:30

HC act2 #3 6:20
HC act2 #4 7:50
HC act2 #5 9:40

MS act1 #6 12:35
MS act1 #7 13:00
MS act1 #8 14:30
MS act1 #9 16:20

MS act2 #10 18:00
MS act2 #11 19:00
MS act2 #12 20:05

OO act1 #13 22:30
OO act1 #14 24:10

OO act2 #15 27:10
OO act2 #16 28:20

LR act1 #17 33:58
LR act1 #18 35:50

LR act2 #19 37:55
LR act2 #20 39:40
LR act2 #21 41:35
LR act2 #22 42:50

MM act1 #23 47:15
MM act1 #24 48:45

MM act2 #25 54:27
MM act2 #26 1:00:00

TM act1 #27 1:03:55
TM act1 #28 1:05:33

TM act2 #29 1:10:10

Note that I had to restart both Mirage Saloon act1 & Lava Reef act2 after collecting a special stage ring because you can't backtrack to get some of the other special stage rings in those acts.

Hopefully this is a useful guide for anyone looking for special stage rings in Encore Mode!