Sooner or later my warlock will need to probe minds again with Detect Thoughts and will end up matching wits with
a Sicilian the target in a contested intelligence check.
Either way, the target knows that you are probing into its mind, and unless you shift your attention to another creature's thoughts, the creature can use its action on its turn to make an Intelligence check contested by your Intelligence check; if it succeeds, the spell ends.
— Detect Thoughts
My Pact of the Tome Warlock, her patron is a Great Old One, only has a 12 intelligence, and we play a low magic item game (we haven't had any at level 4 — it's low combat so that hasn't been an issue). I have concentration spells that would help (i.e. Cause Fear, Hex, Guidance), but I can't use as Detect Thoughts is also concentration. I have just learnt Find Familiar, but I'd want to check with my DM before assuming the help action from the familiar would be allowed.
How can increase my chances to win the check? Preference is given to solo solutions; roleplay-wise my character's warlock abilities are being treated as something the other characters don't want to get too involved in.