diff --git a/share/spack/gitlab/cloud_pipelines/stacks/e4s/spack.yaml b/share/spack/gitlab/cloud_pipelines/stacks/e4s/spack.yaml
index d2ed08d04d..fe054d3345 100644
--- a/share/spack/gitlab/cloud_pipelines/stacks/e4s/spack.yaml
+++ b/share/spack/gitlab/cloud_pipelines/stacks/e4s/spack.yaml
@@ -77,7 +77,9 @@ spack:
     - sundials +cuda cuda_arch=70
     - superlu-dist +cuda cuda_arch=70
     - tasmanian +cuda cuda_arch=70
-    - trilinos@13.2.0 +cuda cuda_arch=70
+    # Trilinos: enable CUDA, Kokkos, and important Tpetra-era solver packages;
+    # disable Epetra; disable ETI to speed up CI; disable all other TPLs
+    - trilinos@13.2.0 +cuda cuda_arch=70 +wrapper    +amesos2 +belos +ifpack2 +kokkos +muelu +nox +stratimikos +tpetra    ~amesos ~anasazi ~aztec ~epetraext ~ifpack ~isorropia ~ml ~teko ~tempus ~zoltan ~zoltan2     ~explicit_template_instantiation ~adios2~basker~boost~chaco~complex~debug~dtk~epetraextbtf~epetraextexperimental~epetraextgraphreorderings~exodus~float~fortran~gtest~hypre~intrepid~intrepid2~ipo~mesquite~minitensor~mumps~openmp~phalanx~piro~rocm~rol~rythmos~sacado~scorec~shards~shared~shylu~stk~stokhos~strumpack~suite-sparse~superlu~superlu-dist~trilinoscouplings~x11
     - umpire ~shared +cuda cuda_arch=70
     - vtk-m +cuda cuda_arch=70
     - zfp +cuda cuda_arch=70
diff --git a/var/spack/repos/builtin/packages/trilinos/package.py b/var/spack/repos/builtin/packages/trilinos/package.py
index 88c6714fc5..a130447979 100644
--- a/var/spack/repos/builtin/packages/trilinos/package.py
+++ b/var/spack/repos/builtin/packages/trilinos/package.py
@@ -174,7 +174,7 @@ class Trilinos(CMakePackage, CudaPackage, ROCmPackage):
 
     # ###################### Conflicts ##########################
 
-    # Epetra packages
+    # Epetra stack
     with when('~epetra'):
         conflicts('+amesos')
         conflicts('+aztec')
@@ -188,8 +188,11 @@ class Trilinos(CMakePackage, CudaPackage, ROCmPackage):
         conflicts('+epetraextbtf')
         conflicts('+epetraextexperimental')
         conflicts('+epetraextgraphreorderings')
+    with when('+teko'):
+        conflicts('~stratimikos')
+        conflicts('@:12 gotype=long')
 
-    # Tpetra packages
+    # Tpetra stack
     with when('~kokkos'):
         conflicts('+cuda')
         conflicts('+rocm')
@@ -204,31 +207,29 @@ class Trilinos(CMakePackage, CudaPackage, ROCmPackage):
         conflicts('+teko')
         conflicts('+zoltan2')
 
-    with when('+teko'):
-        conflicts('~amesos')
-        conflicts('~anasazi')
-        conflicts('~aztec')
-        conflicts('~ifpack')
-        conflicts('~ml')
-        conflicts('~stratimikos')
-        conflicts('@:12 gotype=long')
+    with when('~zoltan'):
+        conflicts('+isorropia')
+        conflicts('+scorec')
+        conflicts('+shylu')
+        conflicts('+zoltan2')
+    with when('~shards'):
+        conflicts('+intrepid')
+        conflicts('+intrepid2')
+        conflicts('+scorec')
+        conflicts('+stk')
+    with when('+scorec'):
+        conflicts('~mpi')
+        conflicts('~stk')
 
     # Known requirements from tribits dependencies
     conflicts('+aztec', when='~fortran')
     conflicts('+basker', when='~amesos2')
-    conflicts('+minitensor', when='~boost')
     conflicts('+ifpack2', when='~belos')
     conflicts('+intrepid', when='~sacado')
-    conflicts('+intrepid', when='~shards')
-    conflicts('+intrepid2', when='~shards')
-    conflicts('+isorropia', when='~zoltan')
+    conflicts('+minitensor', when='~boost')
     conflicts('+phalanx', when='~sacado')
-    conflicts('+scorec', when='~mpi')
-    conflicts('+scorec', when='~shards')
-    conflicts('+scorec', when='~stk')
-    conflicts('+scorec', when='~zoltan')
+    conflicts('+stokhos', when='~kokkos')
     conflicts('+tempus', when='~nox')
-    conflicts('+zoltan2', when='~zoltan')
 
     # Only allow DTK with Trilinos 12.14, 12.18
     conflicts('+dtk', when='~boost')
@@ -279,12 +280,8 @@ class Trilinos(CMakePackage, CudaPackage, ROCmPackage):
 
     # Old trilinos fails with new CUDA (see #27180)
     conflicts('@:13.0.1 +cuda', when='^cuda@11:')
-
-    # See discussion on the PR adding cuda@11.6 for details.
-    #
-    #     https://github.com/spack/spack/pull/28439
-    #
-    conflicts('+cuda', when='^cuda@11.6.0:')
+    # Build hangs with CUDA 11.6 (see #28439)
+    conflicts('+cuda +stokhos', when='^cuda@11.6:')
 
     # stokhos fails on xl/xl_r
     conflicts('+stokhos', when='%xl')