Get thrown exception in finally block

Is there a way, how to get currently thrown exception (if exists)?

I would like reduce amount of code and apply some reuse for task looks like:

Exception thrownException = null;
try {
    // some code with 3rd party classes, which can throw unexpected exceptions
}
catch( Exception exc ) {
    thrownException = exc;
    LogException( exc );
}
finally {
    if ( null == thrownException ) {
        // some code
    }
    else {
        // some code
    }
}

and replace it with this code:

using( ExceptionHelper.LogException() ) {
    // some code with 3rd party classes, which can throw unexpected exceptions
}
using( new ExceptionHelper { ExceptionAction = ()=> /*some cleaning code*/ } ) {
    // some code with 3rd party classes, which can throw unexpected exceptions
}

public class ExceptiohHelper : IDisposable {
    public static ExceptionHelper LogException() {
        return new ExceptionHelper();
    }

    public Action SuccessfulAction {get; set;}
    public Action ExceptionAction {get; set;}

    public void Dispose() {
        Action action;
        Exception thrownException = TheMethodIDontKnow();
        if ( null != thrownException ) {
            LogException( thrownException );
            action = this.ExceptionAction;
        }
        else {
            action = this.SuccessfulAction;
        }

        if ( null != action ) {
            action();
        }
    }
}

Is this scenario posible?

Thanks