I normaly carry a small Fiskers saw and pruning shears in my hydration pack. That is unless I plan on going lighter and not stopping anywhere. If I know of larger logs like over 7 inches or so I take a Silke saw. It is the Silky Professional BIGBOY 2000 Folding Saw 360mm XL Teeth (356-36)Silky Professional BIGBOY 2000 Folding Saw 360mm XL Teeth (356-36). This is too large for my hydration pack. I have these rubber strap blocks I use to strap the saw to my bars. It is a carbon bar so I wrap the saw in a towel and make sure the exposed blade is not towards the bar or me.
GUP (GÜP) Industries Holster Utility Strap, One Size, Self Fastener Bicycle Canister Carry Solution With Non-Slip Rubber Grommet To Protect Bike Frame From Canister Multi-use hook loop strap closure
https://a.co/d/7m6TQsQ
Oh also wrap the Fiskers saw in a small dust rag so it does not wear holes in the hydration pack. The pack has a large storage pocket.