Not really - the initial base needs to be attached to a substrate (e.g. CPG) or support. It's only removed from this substrate at the final step once you have finished extending the chain.

Think of the support as the thing that keeps your DNA in the bottle when you're doing the chemistry; during the synthesis you're repeatedly adding chemicals then washing them away.

If your DNA product isn't firmly attached to something during this process, it's going to get washed away too.

It's conceivable you could remove the DNA from the support, capture it, amplify it with PCR, then reattach to support to continue the extension, but the re-attach part would be very difficult - you'd be dealing with a long floppy chain and trying to attach one end to a solid anchor via some unknown complex chemistry. There would be side-products, loops, breaks, etc. to deal with. Someone has probably tried it, but it's not something I've ever encountered.