Because there is no way to get a tree to drop anything that isn't a log, your only recourse is to remove the carpenter's shop from your civilization's entity tokens and create a custom workshop.
[BUILDING_WORKSHOP:WOODSHOP]
[NAME:Woodshop]
[NAME_COLOR:6:0:1]
[DIM:3:3]
[WORK_LOCATION:2:2]
[BUILD_LABOR:CARPENTER]
[BUILD_KEY:CUSTOM_SHIFT_R]
[BLOCK:1:0:0:0]
[BLOCK:2:0:0:0]
[BLOCK:3:0:0:0]
[TILE:0:1:' ':' ':150]
[TILE:0:2:' ':' ':'/']
[TILE:0:3:'-':' ':' ']
[COLOR:0:1:0:0:0:0:0:0:6:0:0]
[COLOR:0:2:0:0:0:0:0:0:6:0:0]
[COLOR:0:3:6:0:0:0:0:0:0:0:0]
[TILE:1:1:' ':' ':'=']
[TILE:1:2:'-':' ':8]
[TILE:1:3:' ':' ':150]
[COLOR:1:1:0:0:0:0:0:0:6:0:0]
[COLOR:1:2:6:0:0:0:0:0:6:0:0]
[COLOR:1:3:0:0:0:0:0:0:6:0:0]
[TILE:2:1:'-':' ':8]
[TILE:2:2:' ':' ':8]
[TILE:2:3:' ':150:' ']
[COLOR:2:1:6:0:0:0:0:0:6:0:0]
[COLOR:2:2:0:0:0:0:0:0:6:0:0]
[COLOR:2:3:0:0:0:6:0:0:0:0:0]
[TILE:3:1:150:' ':8]
[TILE:3:2:' ':' ':8]
[TILE:3:3:' ':240:' ']
[COLOR:3:1:6:0:0:0:0:0:6:7:0]
[COLOR:3:2:0:0:0:0:0:0:6:7:0]
[COLOR:3:3:0:0:0:7:0:1:0:0:0]
[BUILD_ITEM:1:NONE:NONE:NONE:NONE][BUILDMAT][WORTHLESS_STONE_ONLY][CAN_USE_ARTIFACT]
Since your idea is to get more wood from fewer trees, you want a process that turns logs into boards. You can make boards using the tool entry.
[ITEM_TOOL:ITEM_TOOL_BOARDS]
[NAME:board:boards]
[VALUE:1]
[HARD_MAT]
[TILE:22]
[SIZE:2000]
[MATERIAL_SIZE:1]
You can use your new building to create these tools.
[REACTION:SAW_BOARDS]
[NAME:saw logs into boards]
[BUILDING:WOODSHOP:CUSTOM_B]
[REAGENT:A:1:WOOD:NONE:NONE:NONE]
[PRODUCT:100:4:TOOL:ITEM_TOOL_BOARDS:GET_MATERIAL_FROM_REAGENT:A:NONE]
[SKILL:CARPENTER]
While you're at it, go ahead and make a reaction to create wooden blocks the same way.
[REACTION:SAW_BLOCKS]
[NAME:saw logs into blocks]
[BUILDING:WOODSHOP:CUSTOM_B]
[REAGENT:A:1:WOOD:NONE:NONE:NONE]
[PRODUCT:100:4:BLOCK:NONE:GET_MATERIAL_FROM_REAGENT:A:NONE]
[SKILL:CARPENTER]
And finally, go back and re-enter all of the jobs a carpenter's shop is normally responsible for.
[REACTION:MAKE_BEDS]
[NAME:make wooden beds]
[BUILDING:WOODSHOP:CUSTOM_SHIFT_B]
[REAGENT:A:1:TOOL:ITEM_TOOL_BOARDS:NONE:NONE]
[PRODUCT:100:1:BED:NONE:GET_MATERIAL_FROM_REAGENT:A:NONE]
[SKILL:CARPENTER]
Once you've gone through and made nest boxes, bins, barrels, buckets, hatch covers, doors, and a terrifyingly long array of other items, you will have a carpentry shop that essentially gives you four boards for every log. You can go on to do the same thing to the craftsman's workshop and bowyer's shop, but it will be fine if you decide not to. You can even make the sawing logs and blocks thing give you branches, which are a tool you could use for arrows, bows, and smaller crafts.
So now you have a carpenter's shop that will really stretch wood production. What are some other ways you can ease the burden? Try making a wheat plant that can be milled into flour or processed into a tool called straw hay, and give the craftsman's workshop a reaction to make straw pallets instead of beds. You could do also do a reaction using cloth and feathers or straw to create a divan, which would work like a bed.
There may be a few bugs in the raws I posted here. I didn't bother testing them.
It's also a good idea to build houses by channeling out a bottom floor and building a wooden floor a z-level above it to create a basement. When you start the house this way, you can slap several down relatively quickly and stick beds down there while the rest of the structure is built. Once the family moves upstairs, you can use the basement for food storage etc. You can also save time by making longhouses and inns that house ten or more individuals at once. You can even do a sort of mead-hall at the beginning, where everyone lives communally until you have enough labor and supplies for the commoners to settle away from your ever-expanding palace.