I have a table like
CONTACT
Name Null Type
It isn't stored as an exponential, it's stored in Oracle's internal number representation. When you query it, it's displayed with your sessions's current number format:
select 0.0000000555559080767 from dual;
0.0000000555559080767
---------------------------------------
5.6E-08
Which you can override in SQL*Plus or SQL Developer:
set numformat 999.9999999999999999999
select 0.0000000555559080767 from dual;
0.0000000555559080767
------------------------
.0000000555559080767
Or explicitly format the value as a string, for display only:
set numf ""
select to_char(0.0000000555559080767, '9990.9999999999999999999') from dual;
TO_CHAR(0.000000055555908
-------------------------
0.0000000555559080767
If you have a client application retrieving and using the value then it should query it as a float, into a suitable data type for the client's language, and then it's up to the client how it's displayed.
You also shouldn't be inserting a string into the float column, that just does an implicit conversion; the last argument in your insert should be 0.0000000555559080767 rather than the quoted '0.0000000555559080767'.