Install sqlite latest version on Centos6.7
I tried this locally just now and it's work, now I have sqlite3.10.0 installed on centos6.7 :
Download last version of sqlite from : http://www.sqlite.org/download.html.
Or on the server using : wget http://www.sqlite.org/2016/sqlite-autoconf-3100000.tar.gz.
unzip it : tar xvfz sqlite-autoconf-3100000.tar.gz
cd sqlite-autoconf-3100000
./configure
make
make install
last step if you're using sqlite3 in php, you should recompile it.
Hope this will help you.
Correct way will be to rebuild rpm with never version, but best way is to have it as separate package to maintain compatibility with old version: https://serverfault.com/a/399036/102979