When NIntegrate doesn't issue errors, and using higher precision converges, than the most likely issue is that Integrate couldn't figure out how to handle branch cut issues correctly. One trick that I like to use in such cases is to replace a problematic parameter with a constant, integrate, and then replace the constant with the parameter after the integration. This approach works for your example:
int = Integrate[
Exp[I EulerGamma t]/Sqrt[t^2 - t0^2] t^(-2(n+1)),
{t, t0, Infinity}
];
int //TeXForm
$\frac{1}{2} \pi G_{1,3}^{2,0}\left(\frac{\gamma ^2}{4}|
\begin{array}{c}
\frac{33}{2} \\
0,16,\frac{1}{2} \\
\end{array}
\right)+\frac{i \gamma \pi \left(-1187907847944849195179530749609375 \gamma \pi
\pmb{H}_0(\gamma )+204811697921525723306815646484375 \gamma ^3 \pi \pmb{H}_0(\gamma
)-10619865818153185652945996484375 \gamma ^5 \pi \pmb{H}_0(\gamma
)+262968105973316978072948484375 \gamma ^7 \pi \pmb{H}_0(\gamma
)-3811131970627782290912296875 \gamma ^9 \pi \pmb{H}_0(\gamma )+36296494958359831342021875
\gamma ^{11} \pi \pmb{H}_0(\gamma )-244915620501753247921875 \gamma ^{13} \pi
\pmb{H}_0(\gamma )+1234868674798755871875 \gamma ^{15} \pi \pmb{H}_0(\gamma
)-4842622254112768125 \gamma ^{17} \pi \pmb{H}_0(\gamma )+15248922977413125 \gamma ^{19} \pi
\pmb{H}_0(\gamma )-39607592149125 \gamma ^{21} \pi \pmb{H}_0(\gamma )+86973193125 \gamma
^{23} \pi \pmb{H}_0(\gamma )-165663225 \gamma ^{25} \pi \pmb{H}_0(\gamma )+283185 \gamma
^{27} \pi \pmb{H}_0(\gamma )-465 \gamma ^{29} \pi \pmb{H}_0(\gamma )+\gamma ^{31} \pi
\pmb{H}_0(\gamma )+3672223350604861716931018752000000-439555539574028925954719543981250
\gamma ^2+21860810199430579221603421162500 \gamma ^4-534364371662614944284197113750 \gamma
^6+7699969665250582145431005000 \gamma ^8-73107664089026909823731250 \gamma
^{10}+492396088595579434129500 \gamma ^{12}-2479717198178026269750 \gamma
^{14}+9716498724447673200 \gamma ^{16}-30578708495906550 \gamma ^{18}+79392252095100 \gamma
^{20}-174283041330 \gamma ^{22}+331901640 \gamma ^{24}-567318 \gamma ^{26}+932 \gamma ^{28}-2
\gamma ^{30}\right) J_1(\gamma )}{32891354616711691270902251520000000}-\frac{i \gamma ^2 \pi
\left(\gamma ^{28} (928-465 \pi \pmb{H}_1(\gamma ))+\gamma ^{30} (\pi \pmb{H}_1(\gamma
)-2)+9 \gamma ^{26} (31465 \pi \pmb{H}_1(\gamma )-62826)-225 \gamma ^{24} (736281 \pi
\pmb{H}_1(\gamma )-1470032)-61425 \gamma ^{20} (644812245 \pi \pmb{H}_1(\gamma
)-1286776016)+45 \gamma ^{22} (1932737625 \pi \pmb{H}_1(\gamma )-3858073682)+326025 \gamma
^{18} (46772250525 \pi \pmb{H}_1(\gamma )-93299878898)-20539575 \gamma ^{16} (235770324075
\pi \pmb{H}_1(\gamma )-470043834848)+40186125 \gamma ^{14} (30728732237775 \pi
\pmb{H}_1(\gamma )-61214081542046)-3656937375 \gamma ^{12} (66972877954125 \pi
\pmb{H}_1(\gamma )-133262038442432)+201131555625 \gamma ^{10} (180461463869115 \pi
\pmb{H}_1(\gamma )-358448255769854)-19912024006875 \gamma ^8 (191398522285425 \pi
\pmb{H}_1(\gamma )-379070999744432)+139384168048125 \gamma ^6 (1886642576813475 \pi
\pmb{H}_1(\gamma )-3716840811465638)-407698691540765625 (2913690606794775 \pi
\pmb{H}_1(\gamma )-4503599627370496)+27179912769384375 \gamma ^2 (7535406741710625 \pi
\pmb{H}_1(\gamma )-14210240345493274)-1294281560446875 \gamma ^4 (8205220674307125 \pi
\pmb{H}_1(\gamma )-15982961194617584)\right) J_0(\gamma
)}{32891354616711691270902251520000000}+\frac{9694845 i \gamma \left(1-\frac{5 \gamma
^2}{29}+\frac{7 \gamma ^4}{783}-\frac{13 \gamma ^6}{58725}+\frac{13 \gamma
^8}{4052025}-\frac{13 \gamma ^{10}}{425462625}+\frac{\gamma ^{12}}{4850273925}-\frac{\gamma
^{14}}{961970995125}+\frac{\gamma ^{16}}{245302603756875}-\frac{\gamma
^{18}}{77901098307361875}+\frac{\gamma ^{20}}{29991922848334321875}-\frac{\gamma
^{22}}{13658321665131450181875}+\frac{\gamma ^{24}}{7170618874194011345484375}-\frac{\gamma
^{26}}{4194812041403496637108359375}+\frac{\gamma
^{28}}{2554640533214729451998990859375}-\frac{\gamma
^{30}}{1187907847944849195179530749609375}\right) \pi }{134217728}$
Assuming that all instances of EulerGamma come from my insertion of EulerGamma into the integral, we can do:
N[int /. EulerGamma -> w, 30]
0.0597416653055238422640336356729 + 0.0963766277967734573617915628157 I
which reproduces the NIntegrate result.
