Maybe my answer is a little bit late, but you can do it by either
- ContinuationInterceptor
- CopyableThreadContextElement
More elaborated answer with examples is provided here.
Below I will show an example for ContinuationInterceptor copied from those answers.
class WrappedDispatcher(
    private val dispatcher: ContinuationInterceptor,
    private var savedCounter: Int = counterThreadLocal.get() ?: 0
) : AbstractCoroutineContextElement(ContinuationInterceptor), ContinuationInterceptor {
    override fun <T> interceptContinuation(continuation: Continuation<T>): Continuation<T> =
        dispatcher.interceptContinuation(ContinuationWrapper(continuation))
    private inner class ContinuationWrapper<T>(val base: Continuation<T>) : Continuation<T> by base {
        override fun resumeWith(result: Result<T>) {
            counterThreadLocal.set(savedCounter)
            try {
                base.resumeWith(result)
            } finally {
                savedCounter = counterThreadLocal.get()
            }
        }
    }
}
and usage
val counterThreadLocal: ThreadLocal<Int> = ThreadLocal()
fun showCounter() {
    println("-------------------------------------------------")
    println("Thread: ${Thread.currentThread().name}\n Counter value: ${counterThreadLocal.get()}")
}
fun main() {
    runBlocking(WrappedDispatcher(Dispatchers.IO)) {
        showCounter()
        counterThreadLocal.set(2)
        delay(100)
        showCounter()
        counterThreadLocal.set(3)
        withContext(WrappedDispatcher(Dispatchers.Default)) {
            println("__________NESTED START___________")
            counterThreadLocal.set(4)
            showCounter()
            println("__________NESTED END_____________")
        }
        delay(100)
        showCounter()
    }
}
output will be
-------------------------------------------------
Thread: DefaultDispatcher-worker-1
 Counter value: 0
-------------------------------------------------
Thread: DefaultDispatcher-worker-1
 Counter value: 2
__________NESTED START___________
-------------------------------------------------
Thread: DefaultDispatcher-worker-3
 Counter value: 4
__________NESTED END_____________
-------------------------------------------------
Thread: DefaultDispatcher-worker-3
 Counter value: 3