Hi Ivan,
I would like to point out few important things about Oracle client update,
1. Create soft link to point to new version (As Marius Burger suggested)
2. Set proper environment variable for <sid>adm user (As RB suggested)
Why because files .dbenv.csh and .dbenv.sh in your <sid>adm home directory use a line “ADD=/oracle/client/10x_64/instantclient”
If you upgrade the Oracle client to 11X then you need to update those files(.dbenv.csh and .dbenv.sh) as well.
Restart of SAP system need after Oracle client upgrade .
According to the error you are facing now, Please follow as like of Marius Burger and RB suggested 
Regarding , KK