Boot the router normally, then transfer the firmware over to the router (/tmp is good) somehow. You can SCP from your computer, or if your Internet connection is up from the router you can use wget to fetch it, e.g.
root@OpenWrt:/# cd /tmp root@OpenWrt:/tmp# wget http://downloads.openwrt.org/snapshots/trunk/mvebu/generic/openwrt-mvebu-armada-xp-mamba-squashfs-sysupgrade.tar
then either of
root@OpenWrt:/tmp# sysupgrade -v openwrt-mvebu-armada-xp-mamba-squashfs-sysupgrade.tar
Don't save settings:
root@OpenWrt:/tmp# sysupgrade -n -v openwrt-mvebu-armada-xp-mamba-squashfs-sysupgrade.tar
If you want to go to Linksys or something else it's the same syntax; just replace the sysupgrade.tar file with the .img or .bin or whatever it uses.
n.b., in case you see it elsewhere: with most routers you can use sysupgrade with http directly (e.g. sysupgrade -v http://whatever/blah.bin), but that didn't seem to work with a .tar file last I tried.