You may be thinking of the airside transfer buses. These aren’t an option for you because you won’t be able to enter T4 airside with a T3 boarding pass. The only viable transfer options for you the following morning are those that particularly stut has laid out.

I guess it might be possible to do that transfer on arrival provided no-one asks for proof of departure from T4 before you get on the bus, but there’s no real advantage - the wait for the transfer buses can be lengthy, and you’d arguably have a more complex exit from T4 through immigration and customs once you got there.