"can already start" is correct and usable in written English.
You can use "can already start" when you want to emphasize that something can begin shortly after the present. For example, "Now that the paperwork has been filled out, we can already start the project."
The West Indies, meanwhile, can already start packing.
"We want to show investors they can already start exposing themselves to that market because a larger pool of climate change bonds exists.
Another method of compensating for the shorter holiday season this year — six fewer days compared with last — is to tell customers they "can already start shopping and get some very good deals between now and Thanksgiving," Mr. Pearson said.
Even if politics and funding bodies still need some time to move, individual institutions can already start setting good examples.
… It probably won't happen overnight, but we can already start to make very positive changes now".
And developers can already start playing around with Project Things and contribute to development.
But if you're determined to own a home in New York, you can already get started on an acquisition strategy based on my friend's success.
We will continue tomorrow on the second day in Kharkiv with camera introductions and more brainstorming before we can hopefully already start shooting the first films of the workshop in the afternoon.
You can already begin to see this starting with the G.O.P. on marriage equality, which will probably have a pro-gay marriage nominee at some point in the mid-2020s and might have a pro-civil unions nominee in 2016.
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If it should be clear already earlier that the prerequisites for performing these experiments are fulfilled, this second sequence of tests can already be started earlier on in the mission.
