How to tweak a Lynksys Wireless Router to reach further?

Bigger antenna, more DBI, netstumbler.

You can boost the output power for Linksys and other similar routers using 3rd-party firmware such as DD-WRT. The v5 has special needs, but AFAIK, it can still be done.

Note however that this is like overclocking, and you can decrease the life of the device by exceeding the normal output power significantly, and that better antennae are generally preferred to more power output. There are also some parabolic reflector hacks that can be used with existing antennae.

In addition, the router/AP is only one side of the problem; you're likely to get some improvement by having a more sensitive / better transmitting client.

MIMO devices are advertised as having better reflection and obstruction tolerance.

Some WRT54G v5's are thought to be lemons.

DD-WRT / other 3rd-party firmware + another compatible wireless router gives you the option to create a wireless bridge, with clients being connected through a wire through the second wireless router which is configured as a wireless client bridge. This option in theory gets you better radio performance, antennae options and output increasing capability on both ends. But you might still be better off with higher-tech devices, e.g. MIMO, on both ends.

You can also do WDS wireless repeating using DD-WRT / etc., and another compatible wireless router. This is the traditional way of extending range, but it chews up some of the bandwidth in the process.

Before the advent of wireless, all networking had to be wired, so this can be a viable solution, and the rewards for running wire include much better performance, reliability, and security. Wired gigabit >> wireless.