Any good practise design pattern fror Exception handling

I have exception handling code in every method for the below code for the bottom level methods

throw new Exception("The error that happens");

Is there any way I can avoid writing this code again and again in each method?

I am trying to write my own code and not using any log frameworks

private void TopLevelMethod()
{
    try
    {
        SomeMethod();
    }
    catch (Exception ex)
    {
        // Log/report exception/display to user etc.
    }
}

private void SomeMethod()
{
    TestPartA();
    TestPartB();
    TestPartC();
    TestPartD();
}

private void TestPartA()
{
    // Do some testing...
    try
    {
        if (somethingBadHappens)
        {
            throw new Exception("The error that happens");
        }
    }
    catch (Exception)
    {
        // Cleanup here. If no cleanup is possible, 
        // do not catch the exception here, i.e., 
        // try...catch would not be necessary in this method.

        // Re-throw the original exception.
        throw;
    }
}

private void TestPartB()
{
    // No need for try...catch because we can't do any cleanup for this method.
    if (somethingshappens)
    {
        throw new Exception("The error that happens");
    }
}