diff --git a/tokio/src/task/local.rs b/tokio/src/task/local.rs
index 7334fbeda64..587c45694a5 100644
--- a/tokio/src/task/local.rs
+++ b/tokio/src/task/local.rs
@@ -4,7 +4,7 @@ use crate::runtime::task::{self, JoinHandle, LocalOwnedTasks, Task};
 use crate::sync::AtomicWaker;
 use crate::util::VecDequeCell;
 
-use std::cell::{Cell, RefCell};
+use std::cell::Cell;
 use std::collections::VecDeque;
 use std::fmt;
 use std::future::Future;
@@ -253,7 +253,7 @@ pin_project! {
     }
 }
 
-thread_local!(static CURRENT: RefCell<Option<Rc<Context>>> = RefCell::new(None));
+thread_local!(static CURRENT: Cell<Option<Rc<Context>>> = Cell::new(None));
 
 cfg_rt! {
     /// Spawns a `!Send` future on the local task set.
@@ -303,7 +303,8 @@ cfg_rt! {
           F::Output: 'static
     {
         CURRENT.with(|maybe_cx| {
-            match maybe_cx.borrow().as_ref() {
+            let ctx = clone_rc(maybe_cx);
+            match ctx {
                 None => panic!("`spawn_local` called from outside of a `task::LocalSet`"),
                 Some(cx) => cx.spawn(future, name)
             }
@@ -328,7 +329,6 @@ pub struct LocalEnterGuard(Option<Rc<Context>>);
 impl Drop for LocalEnterGuard {
     fn drop(&mut self) {
         CURRENT.with(|ctx| {
-            // *ctx.borrow_mut() = self.0.take();
             ctx.replace(self.0.take());
         })
     }
@@ -354,7 +354,7 @@ impl LocalSet {
     /// Enter current LocalSet context
     pub fn enter(&self) -> LocalEnterGuard {
         CURRENT.with(|ctx| {
-            let old = ctx.borrow_mut().replace(self.context.clone());
+            let old = ctx.replace(Some(self.context.clone()));
             LocalEnterGuard(old)
         })
     }
@@ -586,10 +586,9 @@ impl LocalSet {
     }
 
     fn with<T>(&self, f: impl FnOnce() -> T) -> T {
-        // CURRENT.set(&self.context, f)
         CURRENT.with(|ctx| {
             struct Reset<'a> {
-                ctx_ref: &'a RefCell<Option<Rc<Context>>>,
+                ctx_ref: &'a Cell<Option<Rc<Context>>>,
                 val: Option<Rc<Context>>,
             }
             impl<'a> Drop for Reset<'a> {
@@ -597,7 +596,7 @@ impl LocalSet {
                     self.ctx_ref.replace(self.val.take());
                 }
             }
-            let old = ctx.borrow_mut().replace(self.context.clone());
+            let old = ctx.replace(Some(self.context.clone()));
 
             let _reset = Reset {
                 ctx_ref: ctx,
@@ -725,23 +724,33 @@ impl<T: Future> Future for RunUntil<'_, T> {
     }
 }
 
+fn clone_rc<T>(rc: &Cell<Option<Rc<T>>>) -> Option<Rc<T>> {
+    let value = rc.take();
+    let cloned = value.clone();
+    rc.set(value);
+    cloned
+}
+
 impl Shared {
     /// Schedule the provided task on the scheduler.
     fn schedule(&self, task: task::Notified<Arc<Self>>) {
-        CURRENT.with(|maybe_cx| match maybe_cx.borrow().as_ref() {
-            Some(cx) if cx.shared.ptr_eq(self) => {
-                cx.queue.push_back(task);
-            }
-            _ => {
-                // First check whether the queue is still there (if not, the
-                // LocalSet is dropped). Then push to it if so, and if not,
-                // do nothing.
-                let mut lock = self.queue.lock();
-
-                if let Some(queue) = lock.as_mut() {
-                    queue.push_back(task);
-                    drop(lock);
-                    self.waker.wake();
+        CURRENT.with(|maybe_cx| {
+            let ctx = clone_rc(maybe_cx);
+            match ctx {
+                Some(cx) if cx.shared.ptr_eq(self) => {
+                    cx.queue.push_back(task);
+                }
+                _ => {
+                    // First check whether the queue is still there (if not, the
+                    // LocalSet is dropped). Then push to it if so, and if not,
+                    // do nothing.
+                    let mut lock = self.queue.lock();
+
+                    if let Some(queue) = lock.as_mut() {
+                        queue.push_back(task);
+                        drop(lock);
+                        self.waker.wake();
+                    }
                 }
             }
         });
@@ -754,11 +763,14 @@ impl Shared {
 
 impl task::Schedule for Arc<Shared> {
     fn release(&self, task: &Task<Self>) -> Option<Task<Self>> {
-        CURRENT.with(|maybe_cx| match maybe_cx.borrow().as_ref() {
-            None => panic!("scheduler context missing"),
-            Some(cx) => {
-                assert!(cx.shared.ptr_eq(self));
-                cx.owned.remove(task)
+        CURRENT.with(|maybe_cx| {
+            let ctx = clone_rc(maybe_cx);
+            match ctx {
+                None => panic!("scheduler context missing"),
+                Some(cx) => {
+                    assert!(cx.shared.ptr_eq(self));
+                    cx.owned.remove(task)
+                }
             }
         })
     }