Getting timepart from timestamp
Running SHOW PARAMETERS LIKE 'TIMEZONE'; returns in my case:

key         value               default                
TIMEZONE    America/Los_Angeles America/Los_Angeles 


SELECT CURRENT_TIMESTAMP() results in 2019-11-04 07:48:47.289 -0800.

Then I can alter the session parameter like this ALTER SESSION SET TIMEZONE = 'Europe/Amsterdam';

Now SHOW PARAMETERS LIKE 'TIMEZONE'; gives:

key         value               default 
TIMEZONE    Europe/Amsterdam    America/Los_Angeles


And SELECT CURRENT_TIMESTAMP() results in 2019-11-04 16:49:45.987 +0100

That hopefully shows the logic behind CURRENT_TIMESTAMP.

Answer to actual question: To get the timepart simply use TIMESTAMP::TIME

SELECT 
CURRENT_TIMESTAMP() :: TIME AS Amsterdam_Time,
CONVERT_TIMEZONE('America/Los_Angeles', CURRENT_TIMESTAMP)::TIME AS LA_Time,
CONVERT_TIMEZONE('UTC', CURRENT_TIMESTAMP)::TIME AS UTC_Time;


Results in (with have Amsterdam as default):

AMSTERDAM_TIME  LA_TIME     UTC_TIME
16:55:37        07:55:37    15:55:37