Here are some more zeros of $\zeta^{(k)}(s)$ with $\operatorname{Re}(s) < \frac12$ found with sage:


    k  zero
    2, -0.35508433021047637343 + 3.5908393243989674267 i
    3, -2.1101457926534055013 + 2.5842247720404084808 i
    4, -3.2402527145532669326 + 1.6896108197133965315 i
    4, -0.83754471075524375517 + 3.8476752430859881333 i
    5, -4.2739256609673291219 + 0.66239093925605351714 i
    5, -4.2739256609673291219 - 0.66239093925605351714 i
    5, -2.1841011922856061145 + 3.0795001018135066045 i
    5, 0.28760642074073336717 + 4.6944346849390517355 i
    6, -3.1693828756887856638 + 2.289409321602478671 i
    6, -1.2725578081548747958 + 4.0741784672743930742 i
    7, -3.8750437984692690102 + 1.4917785168218934496 i
    7, -2.3934461266307994251 + 3.4062662320064732901 i
    7, -0.41331687971848942853 + 4.8452970349839958557 i
    8, -4.5682125648555486581 + 0.81115476881064826173 i
    8, -3.2523204085919068777 + 2.7169913803460029554 i
    8, -1.6702731549827933708 + 4.2784414287490737296 i
    8, 0.41829665603240684742 + 5.4752676150632522598 i
    9, -3.945849046353689844 + 2.0451787413259943742 i
    9, -2.6409589700825760293 + 3.6749136820039963426 i
    9, -0.96722009647711882409 + 4.9985369336041978752 i
    10, -4.5121442160084114922 + 1.3320691841545877923 i
    10, -3.4228591767532529916 + 3.0609427521799546739 i
    10, -2.0391435059208024397 + 4.4684432318278930578 i
    10, -0.27483240240270603573 + 5.6133090845225833824 i
    11, -5.0309783500584905982 + 0.76405382608151195607 i
    11, -4.0769374324171281166 + 2.4384470640345412807 i
    11, -2.9061882184131317446 + 3.9131827555009648537 i
    11, -1.4413185933633706999 + 5.1493402952351168812 i
    11, 0.41063992065310427805 + 6.1502251675161887368 i
    12, -4.6217542104762414501 + 1.8307305267522831431 i
    12, -3.6306820396013356257 + 3.3459003484685151686 i
    12, -2.387362966467342712 + 4.6486063663821024758 i
    12, -0.84520925190624343306 + 5.7472734659279260752 i
    13, -5.1019135182813204449 + 1.1817501251741196135 i
    13, -4.2445336021969936603 + 2.7739661902545951775 i
    13, -3.1788394218248838054 + 4.1282947333700902908 i
    13, -1.8653042754449209073 + 5.2971387830938879447 i
    13, -0.24996831630193368259 + 6.2810810441530943019 i