diff --git a/tokio/src/runtime/task/core.rs b/tokio/src/runtime/task/core.rs
index 2b7050a2111..3e07d7c97fd 100644
--- a/tokio/src/runtime/task/core.rs
+++ b/tokio/src/runtime/task/core.rs
@@ -155,7 +155,9 @@ impl<T: Future> CoreStage<T> {
     pub(super) fn with_mut<R>(&self, f: impl FnOnce(*mut Stage<T>) -> R) -> R {
         self.stage.with_mut(f)
     }
+}
 
+impl<T: Future, S: Schedule> Core<T, S> {
     /// Polls the future.
     ///
     /// # Safety
@@ -171,7 +173,7 @@ impl<T: Future> CoreStage<T> {
     /// heap.
     pub(super) fn poll(&self, mut cx: Context<'_>) -> Poll<T::Output> {
         let res = {
-            self.stage.with_mut(|ptr| {
+            self.stage.stage.with_mut(|ptr| {
                 // Safety: The caller ensures mutual exclusion to the field.
                 let future = match unsafe { &mut *ptr } {
                     Stage::Running(future) => future,
@@ -224,7 +226,7 @@ impl<T: Future> CoreStage<T> {
     pub(super) fn take_output(&self) -> super::Result<T::Output> {
         use std::mem;
 
-        self.stage.with_mut(|ptr| {
+        self.stage.stage.with_mut(|ptr| {
             // Safety:: the caller ensures mutual exclusion to the field.
             match mem::replace(unsafe { &mut *ptr }, Stage::Consumed) {
                 Stage::Finished(output) => output,
@@ -234,7 +236,7 @@ impl<T: Future> CoreStage<T> {
     }
 
     unsafe fn set_stage(&self, stage: Stage<T>) {
-        self.stage.with_mut(|ptr| *ptr = stage)
+        self.stage.stage.with_mut(|ptr| *ptr = stage)
     }
 }
 
diff --git a/tokio/src/runtime/task/harness.rs b/tokio/src/runtime/task/harness.rs
index 70415799225..545b01b7ff7 100644
--- a/tokio/src/runtime/task/harness.rs
+++ b/tokio/src/runtime/task/harness.rs
@@ -1,5 +1,5 @@
 use crate::future::Future;
-use crate::runtime::task::core::{Cell, Core, CoreStage, Header, Trailer};
+use crate::runtime::task::core::{Cell, Core, Header, Trailer};
 use crate::runtime::task::state::{Snapshot, State};
 use crate::runtime::task::waker::waker_ref;
 use crate::runtime::task::{JoinError, Notified, Schedule, Task};
@@ -104,13 +104,7 @@ where
                 let header_ptr = self.header_ptr();
                 let waker_ref = waker_ref::<T, S>(&header_ptr);
                 let cx = Context::from_waker(&*waker_ref);
-                let core = self.core();
-                let res = poll_future(
-                    &core.stage,
-                    &self.core().scheduler,
-                    core.task_id.clone(),
-                    cx,
-                );
+                let res = poll_future(self.core(), cx);
 
                 if res == Poll::Ready(()) {
                     // The future completed. Move on to complete the task.
@@ -124,15 +118,13 @@ where
                     TransitionToIdle::Cancelled => {
                         // The transition to idle failed because the task was
                         // cancelled during the poll.
-                        let core = self.core();
-                        cancel_task(&core.stage, core.task_id.clone());
+                        cancel_task(self.core());
                         PollFuture::Complete
                     }
                 }
             }
             TransitionToRunning::Cancelled => {
-                let core = self.core();
-                cancel_task(&core.stage, core.task_id.clone());
+                cancel_task(self.core());
                 PollFuture::Complete
             }
             TransitionToRunning::Failed => PollFuture::Done,
@@ -155,8 +147,7 @@ where
 
         // By transitioning the lifecycle to `Running`, we have permission to
         // drop the future.
-        let core = self.core();
-        cancel_task(&core.stage, core.task_id.clone());
+        cancel_task(self.core());
         self.complete();
     }
 
@@ -190,7 +181,7 @@ where
     /// Read the task output into `dst`.
     pub(super) fn try_read_output(self, dst: &mut Poll<super::Result<T::Output>>, waker: &Waker) {
         if can_read_output(self.header(), self.trailer(), waker) {
-            *dst = Poll::Ready(self.core().stage.take_output());
+            *dst = Poll::Ready(self.core().take_output());
         }
     }
 
@@ -215,7 +206,7 @@ where
             // they are dropping the `JoinHandle`, we assume they are not
             // interested in the panic and swallow it.
             let _ = panic::catch_unwind(panic::AssertUnwindSafe(|| {
-                self.core().stage.drop_future_or_output();
+                self.core().drop_future_or_output();
             }));
         }
 
@@ -325,7 +316,7 @@ where
                 // The `JoinHandle` is not interested in the output of
                 // this task. It is our responsibility to drop the
                 // output.
-                self.core().stage.drop_future_or_output();
+                self.core().drop_future_or_output();
             } else if snapshot.has_join_waker() {
                 // Notify the join handle. The previous transition obtains the
                 // lock on the waker cell.
@@ -457,36 +448,32 @@ enum PollFuture {
 }
 
 /// Cancels the task and store the appropriate error in the stage field.
-fn cancel_task<T: Future>(stage: &CoreStage<T>, id: super::Id) {
+fn cancel_task<T: Future, S: Schedule>(core: &Core<T, S>) {
     // Drop the future from a panic guard.
     let res = panic::catch_unwind(panic::AssertUnwindSafe(|| {
-        stage.drop_future_or_output();
+        core.drop_future_or_output();
     }));
 
+    let id = core.task_id.clone();
     match res {
         Ok(()) => {
-            stage.store_output(Err(JoinError::cancelled(id)));
+            core.store_output(Err(JoinError::cancelled(id)));
         }
         Err(panic) => {
-            stage.store_output(Err(JoinError::panic(id, panic)));
+            core.store_output(Err(JoinError::panic(id, panic)));
         }
     }
 }
 
 /// Polls the future. If the future completes, the output is written to the
 /// stage field.
-fn poll_future<T: Future, S: Schedule>(
-    core: &CoreStage<T>,
-    scheduler: &S,
-    id: super::Id,
-    cx: Context<'_>,
-) -> Poll<()> {
+fn poll_future<T: Future, S: Schedule>(core: &Core<T, S>, cx: Context<'_>) -> Poll<()> {
     // Poll the future.
     let output = panic::catch_unwind(panic::AssertUnwindSafe(|| {
-        struct Guard<'a, T: Future> {
-            core: &'a CoreStage<T>,
+        struct Guard<'a, T: Future, S: Schedule> {
+            core: &'a Core<T, S>,
         }
-        impl<'a, T: Future> Drop for Guard<'a, T> {
+        impl<'a, T: Future, S: Schedule> Drop for Guard<'a, T, S> {
             fn drop(&mut self) {
                 // If the future panics on poll, we drop it inside the panic
                 // guard.
@@ -504,8 +491,8 @@ fn poll_future<T: Future, S: Schedule>(
         Ok(Poll::Pending) => return Poll::Pending,
         Ok(Poll::Ready(output)) => Ok(output),
         Err(panic) => {
-            scheduler.unhandled_panic();
-            Err(JoinError::panic(id, panic))
+            core.scheduler.unhandled_panic();
+            Err(JoinError::panic(core.task_id.clone(), panic))
         }
     };
 
@@ -515,7 +502,7 @@ fn poll_future<T: Future, S: Schedule>(
     }));
 
     if res.is_err() {
-        scheduler.unhandled_panic();
+        core.scheduler.unhandled_panic();
     }
 
     Poll::Ready(())