You may reach this using recursive query
create table days as
with recursive qq as (
select 1 id, 'jan' month, '2021-01-01' date_col union all
select id + 1, substr('janfebmaraprmayjunjulaugsepoctnovdec', 1 + 3 *
strftime('%m', date(date_col, '+1 day')), -3), date(date_col, '+1 day')
from qq
where id <= 364
)
The monstrous line
substr('janfebmaraprmayjunjulaugsepoctnovdec', 1 + 3 * strftime('%m', date(date_col, '+1 day')), -3)
will cut the name of the month you need from the line based on the month number of the date. Ths one is needed because SQLIte seems to be not able to get month name from date.
There were suggestions of how one can get month names from date but it did not worked for me. Try those if you want
Here's an example on dbfiddle