Yes. It won't take much, just a scoop or two distributed throughout. People often take chunky pieces that did not break down completely from the last pile and use them to inoculate the new pile. The compost itself will work just as well.

Yeah
and BTW... I add to my bins continuously, unless and until I need to use some ,I just keep mixing it all in, it breaks down real fast.

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Jon, you know what you're showing us in all those photos of yours, don't you?

It's called 'compost porn'.

Sue

In theory adding finished compost to new material should help get that new material digested faster. However, everything you add to your compost will have the bacteria that are going to digest it and if the right conditions are present they will go to work and digest that material whether some from finished compost is added or not.
Adding finished compost to a new pile has no real advantage if hte mix is near optimal.

Yep, what Kimm said.

If your pile is well-managed and the conditions are optimal and you diligently turn the pile regularly, it probably doesn't make much difference if you add a little finished compost to the new pile.

However, if you're more lax about management and if your pile is fungally dominated and so cold that scallions can grow in it, then mixing in some finished (or partially finished) compost makes a difference.

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Eventually this pile does produce finished compost. This view is from the oldest end - the mushrooms are down by the bin where the newest compost sits.
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Claire (who can't resist lowering the bar after jonhughes has posted his amazing pics)

This is what I mean by fungally dominant. The mushrooms are happily growing on the newest material.

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Claire

WOW Claire ,That is a nice pile, I am definitely lusting ;-)

Ah, Jon, we mortals deal with compost as the will and upper-body strength dictate. My garden and I are happy to wait until the compost is ready enough.

Of course, if someone left a backhoe bagged up on the curbside with a note saying "Free, works fine, take it", I might reconsider.

Claire

Adding some finished compost to a new pile would probably speed up the rate of decomposition initially* but in the medium to long term it would more than likely be irrelevant**.

*Temperature increase to 150F in 20 hours versus 24 hours kind of thing.

**Once the material has reached 150-160F it isn't normally going to go any higher anyways.

Lloyd

Again, the question is whether you want a compost pile that heats up dramatically. Not all composting involves major heat generation. Mother Nature has been carrying out cold composting for millennia.

Claire

Probably won't make any difference in most piles. If you had a pile only of used coffee grounds and shredded paper, it might be a good idea.

But my main 'brown' consists of finely shredded leaves, and I doubt that adding a few scoops of finished compost would speed things up. When I mix up a batch of matched fresh grass and or coffee grounds with finely shredded leaves, the core temp. zooms up to 150 to 160F in two to three days.

Adding a bit of finished compost never hurts. But if you feel your mix is not heating up as much/fast as you expect; the answer may lie with another factor.

Mix of greens and browns
Freshness of greens
Need small pieces
Moisture too wet or dry
Pile too small to retain heat.